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DEU J 47T7-T.25 





A — The Problem of Left and Right 






B — The Problem of Oceans 




C — The Problem of Numbers and Lines 




D — The Problem of the Platypus 




E — The Problem of History 



15— BILL AND I 186 

F — The Problem of Population 

16 — STOP! 201 

17 BUT HOW? 214 


A — The Problem of Left and Right 


I have just gone through a rather unsettling experience. Ordinarily 
it is not veiy difficult to think up a topic for these chapters. 
Some interesting point will occur to me, which will quickly lead 
my mind to a particular line of development, beginning in one 
place and ending in another. Then, I get stalled. 

Today, however, having determined to deal with asymmetry 
(in more than one chapter, veiy likely) and to end with life and 
antilife, I found that two possible stalling points occurred to me. 
Ordinarily, when this happens, one stalling point see ms so much 
superior to me that 1 choose it over the other with a minimum of 

This time, however, the question was whether to stall with 
even numbers or with double refraction, and the arguments raging 
within my head for each case were so equally balanced that 1 
couldn't make up my mind. For two hours I sat at my desk, pon- 
dering first one and then the other and growing steadily un- 

Indeed, I became uncomfortably aware of the resemblance of 
my case to that of "Buridan's ass." 

The reference, here, is to a fourteenth-century French philoso- 
pher, Jean Buridan, who was supposed to have stated the follow- 
ing: "If a hungry ass were placed exactly between two haystacks 
in every respect equal, it would starve to death, because there 
would be no motive why it should go to one rather than to the 

Actually, of course, there's a fallacy here, since the statement 
does not recognize the existence of the random factor. The ass, 
no logician, is bound to turn his head randomly so that one hay- 
stack comes into better view, shuffle his feet randomly so that one 



haystack comes to be closer; and he would end at the haystack 
better seen or more closely approached. 

Which haystack that would be, one could not tell in advance. 
If one had a thousand asses placed exactly between a thousand 
sets of haystack pairs, one could confidently expect that about 
half would turn to the right and half to the left. The individual 
ass, however, would remain unpredictable. 

In the same way, it is impossible to predict whether an honest 
coin, honestly thrown, will come down heads or tails in any one 
particular case, but we can confidently predict that a very large 
number of coins tossed simultaneously (or one coin tossed a very 
large number of times) will show heads just about half the time 
and tails the other half. 

And so it happens that although the chance of the fall of heads 
or tails is exactly even, just fifty-fifty, you can nevertheless call 
upon the aid of randomness to help you make a decision by tossing 
one coin once. 

At this point, I snapped out of my reverie and did what a lesser 
mind would have done two hours before. I tossed a coin. 

Shall we start with even numbers. Gentle Readers? 

I suspect that some prehistoric philosopher must have decided 
that there were two kinds of numbers: peaceful ones and warlike 
ones. The peaceful numbers were those of the type 2, 4, 6, 8, while 
the numbers in between were warlike. 

If there were 8 stone axes and two individuals possessing equal 
claim, it would be easy to hand 4 to each and make peace. If there 
were 7, however, you would have to give 3 to each and then either 
toss away the 1 remaining (a clear loss of a valuable object) or 
let the two disputants fight over it. 

The fact that the original property that marked out the signifi- 
cance of what we now call even and odd numbers was something 
like this is indicated by the very names we give them. 

The word "even" means fundamentally, "flat, smooth, without 
unusual depressions or elevations." We use the word in this sense 



when we say that a person says something "in an even tone of 
voice." An even number of identical coins, for instance, can be 
divided into two piles of exactly the same height. The two piles 
are even in height and hence the number is called even. The even 
number is the one with the property of "equal shares." 

"Odd," on the other hand, is from an old Norse word meaning 
"point" or "tip." If an odd number of coins is divided into two 
piles as nearly equal as possible, one pile is higher by one coin and 
therefore rears a point or tip into the ah', as compared with the 
other. The odd number possesses the property of "unequal 
shares," and it is no accident that the expression "odds" in betting 
implies the wagering of unequal amounts of money by the two 

Since even numbers possess the property of equal shares, they 
were said to have "parity," from a Latin word meaning "equal." 
Originally, this word applied (as logic demanded) to even num- 
bers only, but mathematicians found it convenient to say that if 
two numbers were both even or both odd, they were, in each case 
"of the same parity." An even number and an odd number, 
grouped together, were "of different parity." 

To see the convenience of this convention, consider the fol- 

If two even numbers are added, the sum is invariably even. 
(This can be expressed mathematically by saying that two even 
numbers can be expressed as z,m and zn where m and n are whole 
numbers and that the sum, zm + zn, is still clearly divisible by 
two. However, we are friends, you and I, and I'm sure we can dis- 
pense with mathematical reasoning and that I will find you willing 
to accept my word of honor as a gentleman in such matters. Be- 
sides, you are welcome to search for two even numbers whose 
sum isn't even.) 

If two odd numbers are added, the sum is also invariably even. 

If an odd number and an even number are added, however, the 
sum is invariably odd. 



We can express this more succinctly in symbols, with E standing 
for even and O standing for odd: 

E + E = E 
0 + 0 = E 
E + 0 = 0 
0 + E = 0 

Or, if we are dealing with pail's of numbers only, the concept 
of parity enables us to say it in two statements, rather than four: 

1) Same parities add to even. 

2) Different parities add to odd. 

A veiy similar state of affairs exists with reference to multipli- 
cation, if we divide numbers into two classes: positive numbers 
(+)• and negative numbers ( — ). The product of two positive num- 
bers is invariably positive. The product of two negative numbers 
is invariably positive. The product of a positive and a negative 
number is invariably negative. Using symbols: 

+ x+ = + 

-X- = + 

+ x- = - 
-x + = - 

Or, if we consider all positive numbers as having one kind of 
parity and all negative numbers as another, we can say, in con- 
nection with the multiplication of two numbers: 

1) Same parities multiply to positive. 

2) Different parities multiply to negative. 

The concept of parity — that is, the assignment of all objects 
of a particular class to one of two subclasses and then finding two 
opposing results when objects of the same or of different sub- 
classes are manipulated — can be applied to physical phenomena. 

For instance, all electrically charged particles can be divided 
into two classes: positively charged and negatively charged. Again, 



all magnets possess two points of concentrated magnetism of 
opposite properties: a north pole and a south pole. Let's symbolize 
these as +, — , N and S. 

It turns out that: 

+ and + or N and N = repulsion 

— and — or S and S = repulsion 

+ and — or N and S = attraction 

— and + or S and N = attraction 

Again, we can make two statements: 

1) Like electric charges, or magnetic poles, repel each other. 

2) Opposite electric charges, or magnetic poles, attract each 

The similarity in foim to the situation with respect to the sum- 
ming of odd and even, or the multiplying of positive and negative, 
is obvious. 

When, in any situation, same parities always yield one result 
and different parities always yield the opposite result, we say that 
"parity is conserved." If two even numbers sometimes added up 
to an odd number; or if a positive number multiplied by a nega- 
tive one sometimes yielded a positive product; or if two positively 
charged objects sometimes attracted each other; or if a north 
magnetic pole sometimes repelled a south magnetic pole, we 
would say that, "The law of conservation of parity is violated." 

Certainly in connection with numbers and with electromagne- 
tic phenomena, no one has ever observed the law of conservation 
of parity to have been violated, and no one seriously expects to 
observe a case in the future. 

What about other cases? 

Well, electromagnetism involves a field. That is, any electrically 
charged particle, or any magnet, is surrounded by a volume of 
space within which its properties are made manifest on other 
objects of the same sort. The other objects are also surrounded 
by a volume of space within which their properties are made mani- 
fest on the original object. One speaks, therefore, of an "electro- 


magnetic interaction" involving pairs of objects carrying electric 
charge or magnetic poles. 

Up through the first years of the twentieth century, the only 
other kind of interaction known was the gravitational. 

At first blush, there seems no easy way of involving gravitation 
with parity. There is no way of dividing objects into two groups, 
one with one kind of gravitational property and the other with 
the opposite kind. 

All objects of a given mass possess the same intensity of gravi- 
tational interaction of the same sort. Any two objects with mass 
attract each other. There seems no such thing as "gravitational 
repulsion" (and, according to Einstein's General Theory of Rela- 
tivity there can't be such a thing). It is as though, in gravity, we 
can say only that E + E = Eor + X + = -k 

To be sure, there is a chance that in the field of subatomic 
physics there might be some objects with mass that possess the 
usual gravitational properties and other objects with mass that 
possess gravitational properties of the opposite kind ("antigrav- 
ity"). In that case, the chances are that it would turn out that 
two antigravitational objects attract each other just as two gravi- 
tational objects do; but that an antigravitational and a gravita- 
tional object would repel each other. The situation with respect 
to the gravitational interaction would be the reverse of the elec- 
tromagnetic one (like gravities would attract and unlike gravities 
would repel) but, allowing for that reversal, parity would still 
be conserved. 

The trouble is, though, that the gravitational interaction is so 
much more feeble than the electromagnetic interaction that gravi- 
tational interactions of subatomic particles are as yet impossible 
to measure and a sub-tiny attraction can't be differentiated from 
a sub-tiny repulsion. — So the question of parity and the gravita- 
tional field remains in abeyance. 

As the twentieth century wore on, it came to be recognized that 
the gravitational and electromagnetic interactions were not the 
only ones that existed. Subatomic particles involved something 



else. To be sure, electrons had negative charges and protons had 
positive charges and with respect to this, they behaved in accord- 
ance with the rules of electromagnetic interaction. There were 
other events in the subatomic world, however, that had nothing 
to do with electromagnetism. There was, for instance, some sort 
of interaction involving particles, whether with or without electric 
charge, that showed itself only in the super-close quarters to be 
found within the atomic nucleus. 

Did this "nuclear interaction" involve parity? 

Every subatomic particle has a certain quantum-mechanical 
property which can be expressed in a form involving three quanti- 
ties, x, y, and z. In some cases, it is possible to change the sign of 
all three quantities from positive to negative without changing the 
sign of the expression as a whole. Particles in which this is true 
are said to have "even parity." In other cases, changing the signs 
of the three quantities does change the sign of the entire expres- 
sion and a particle of which this is true is said to have "odd parity." 

Why even and odd? Well, an even-parity particle can break up 
into two even-parity particles or two odd-parity particles, but never 
into one even-parity plus one odd-parity. An odd-parity particle, 
on the other hand, can break up into an odd-parity particle plus 
an even-parity one, but never into two odd-parity particles or two 
even-parity particles. This is analogous to the way in which an 
even number can be the sum of two even numbers or of two odd 
number's, but never the sum of an even number and an odd num- 
ber, while an odd number can be the sum of an even number and 
an odd number, but can never be the sum of two even numbers 
or of two odd numbers. 

But then a particle called the "K-meson" was discovered. It was 
unstable and quickly broke down into "pi-mesons." Some 
K-mesons gave olf two pi-mesons in breaking down and some gave 
df three pi-mesons and that was instantly disturbing. If a K-meson 
did one, it ought not to be able to do the other. Thus an even 
number can be the sum of two odd numbers (10= 3 + 7) 
and an odd number can be the sum of three odd numbers 
(11 = 3 + 7 + 1), but no number can be the sum of two odd 



numbers in one case and three odd numbers in another. It would 
be like expecting a number to be both odd and even. It would, in 
short, represent a violation of the law of conservation of parity. 

Physicists therefore reasoned there must be two kinds of 
K-meson; an even-parity variety ("theta-meson") that broke down 
to two pi-mesons, and an odd-parity variety ("tau-meson") that 
broke down to three pi-mesons. 

This did not turn out to be an altogether satisfactory solution, 
since there seemed to be no possible distinction one could make 
between the theta-meson and the tau-meson except for the num- 
ber of pi-mesons it broke down into. To invent a difference in 
parity for two particles identical in every other respect seemed 
poor practice. 

By 1956, a few physicists had begun to wonder if it weren't pos- 
sible that the law of conservation of parity might not be broken in 
some cases. If that were so, maybe it wouldn't be necessary to try 
to make a distinction between the theta-meson and the tau-meson. 

The suggestion roused the interest of two young Chinese- 
American physicists at Columbia, Chen Ning Yang and Tsung 
Dao Lee, who took into consideration the following — 

There is, as a matter of fact, not one nuclear interaction, but 
two. The one that holds protons and neutrons together within the 
nucleus is an extremely strong one, about 130 times as strong as 
the electromagnetic interaction, so it is called the "strong nu- 
clear interaction." 

There is a second, "weak nuclear interaction" which is only 
about a hundred-trillionth the intensity of the strong nuclear in- 
teraction (but still some trillion-trillion times as intense as the 
unimaginably weak gravitational interaction). 

This meant that there were four types of interaction in the 
universe (and there is some theoretical reason for arguing that a 
fifth of any sort cannot exist, but I would hate to commit myself 
to that): 1) strong nuclear, 2) electromagnetic, 3) weak nuclear, 
and 4) gravitational. 

We can forget about the gravitational interaction for reasons 



I mentioned earlier in the article. Of the other three, it had been 
well established by 1956 that parity was conserved in the strong 
nuclear interaction and in the electromagnetic interaction. Nu- 
merous cases of such conservation were known and the matter 
was considered settled. 

No one, however, had ever systematically checked the weak nu- 
clear interaction with respect to parity, and the breakdown of the 
K-meson involved a weak nuclear interaction. To be sure, all physi- 
cists assumed that parity was conserved in the weak nuclear in- 
teraction but that was only an assumption. 

Yang and Lee published a paper pointing this out — and sug- 
gested experiments that might be performed to check whether the 
weak nuclear interactions conserved parity or not. Those experi- 
ments were quickly earned out and the Yang-Lee suspicion that 
parity would not be conserved was shown to be correct. There 
was very little delay in awarding them shares in the Nobel prize 
in physics in 1957, at which time Yang was thirty-four and Lee, 

You might ask, of course, why parity should be conserved in 
some interactions and not in others — and might not be satisfied 
with the answer "Because that's the way the universe is." 

Indeed, by concentrating too hard on those cases where parity 
is conserved, you might get the notion that it is impossible, in- 
conceivable, unthinkable to deal with a case where it isn't con- 
served. If the conservation of parity is then shown not to hold in 
some cases, the notion arises that this is a tremendous revolution 
that throws the entire structure of science into a state of collapse. 

None of that is so. 

Parity is not so essential a pari of everything that exists that 
it must be conserved in all places, at all times, and under all con- 
ditions. Why shouldn't there be conditions where it isn't con- 
served or, as in the case of gravitational interaction, where it might 
not even apply? 

It is also important to understand that the discovery of the 
fact that parity was not conserved in weak nuclear interactions 



did not "overthrow" the law of conservation of parity, even though 
that was certainly the way in which it was presented in the news- 
papers and even by scientists themselves. The law of conservation 
of parity, in those cases in which its validity had been tested by 
experiment, remained and is still as much in force as ever. 

It was only in connection with the weak nuclear interactions, 
where the validity of the law of conservation of parity had never- 
been tested prior to 1956 and where it had merely been rather 
carelessly assumed that it applied, that there came the change. The 
final experiment merely showed that physicists had made an as- 
sumption they had no real right to make and the law of conserva- 
tion of parity was "overthrown" only where it had never been 
shown to exist in the first place. 

It might help if we look at some familiar, everyday case where 
a law of conservation of parity holds, and then on another where 
it is merely assumed to hold by analogy, but doesn't really. We 
can then see what happened in physics, and why an overthrow of 
something that really isn't there to begin with, improves the struc- 
ture of science and does not damage it. 

Human beings can be divided into two classes: male (M) and 
female (F). Neither two males by themselves nor two females 
by themselves can have children (no C). A male and a female 
together, however, can have children (C). So we can write: 

MandM = noC 
FandF =noC 
M and F = C 
FandM = C 

There is thus the fami li ar parity situation: 

1) Like sexes cannot have children 

2) Opposite sexes can have children. 

To be sure, there are sexually immature individuals, barren fe- 
males, sterile or impotent men, and so on, but these matters are 
details that don't affect the broad situation. As far as the sexes 


and children are concerned, we can say that the human species 
(and, indeed, many other species) conserves parity. 

Because the human species conserves sexual parity with respect 
to childbirth, it is easy to assume it conserves it with respect to 
love and affection as well, so that the feeling arises that sexual 
love ought to exist only between men and women. The fact is, 
though, that parity is not conserved in that respect and that both 
male homosexuality and female homosexuality do exist and have 
always existed. The assumption that parity ought to be conserved 
where, in actual fact, it isn't, has caused many people to find homo- 
sexuality immoral, perverse, abhorrent, and has created oceans of 
woe throughout history. 

Again, in Judeo-Christian culture, the institution of marriage is 
closely associated with childbirth and therefore strictly observes 
the law of conservation of parity that holds for childbirth. A mar- 
riage can take place only between one man and one woman be- 
cause, ideally, that is the simplest system that makes childbirth 

Now, however, there is an increasing understanding that parity, 
which is rigidly conserved with respect to childbirth, is not neces- 
sarily conserved with respect to sexual relations. Increasingly, 
homosexuality is treated not as a sin or a crime, but as, at most, 
a misfortune (if that). 

There is the further attitude, slowly growing in our society, 
that there is no need to force the institution of marriage into the 
tight grip of parity conservation. We hear, more and more fre- 
quently, of homosexual marriages and of group marriages. (The 
old-fashioned institution of polygamy is an example of one kind of 
marriage, enjoyed by many of the esteemed men of the Old Testa- 
ment, in which sexual parity was not conserved.) 

In the next chapter, then, we'll go on with the nature of the 
experiment that established the non-conservation of parity in the 
weak nuclear interaction and consider what happened afterward. 


I received a letter yesterday which criticized my writing style. It 
complained, "you avoid the poetic to the extent that when a cryp- 
tic, glowing, 'charged' phrase occurs to you. I'd be willing to bet 
that you deliberately put it aside and opt for a clearer but more 
pedestrian one." 

All I can say to that is that you bet your sweet life I do. 

As all who read my volumes of science essays must surely be 
aware, 1 have a dislike for the mystical approach to the universe, 
whether in the name of science, philosophy, or religion. 1 also 
have a dislike for the mystical approach to literature. 

I dare say it is possible to evoke an emotional reaction through 
a "cryptic, glowing, 'charged' phrase" but you show me a ciyptic 
phrase and I'll show you any number of readers who, not knowing 
what it means but afraid to admit their ignorance, will say, "My, 
isn't that poetic and emotionally effective." 

Maybe it is, and maybe it isn't; but a vast number of literary 
incompetents get by on the intellectual insecurity of their readers, 
and a vast number of hacks write a vast quantity of bad "poetry" 
and make a living at it. 

For myself, 1 manage to retain a certain amount of intellectual 
security. When 1 read a book that is intended (presumably) for 
the general public and find that I can make neither head nor tail 
of it, it never occurs to me that this is because 1 am lacking in 
intelligence. Rather, 1 reach the calmly assured opinion that the 
author is either a poor writer, a confused thinker, or, most likely, 

Holding these views, it is not surprising that 1 "opt for a clearer 
but more pedestrian" style in my own writing. 

For one thing, my business and my passion (even in my fiction 


writing) is to explain. Partly it is the missionary instinct that 
makes me yearn to make my readers see and understand the uni- 
verse as I see and understand it, so that they may enjoy it as I do. 
Partly, also, I do it because the effort to put things on paper clearly 
enough to make the reader understand, makes it possible for me 
to understand, too. 

I try to teach because whether or not I succeed in teaching 
others, I invariably succeed in teaching myself. 

Yet I must admit that sometimes this self-imposed task of mine 
is harder than other times. Continuing the exposition on parity 
and related topics begun in Chapter 1 is one of the harder times, 
but then no one ever promised me a rose garden, so let's 
continue — 

The conservation laws are the basic generalizations of physics 
and of the physics aspects of all other sciences. In general, a con- 
servation law says that some particular overall measured property 
of a closed system (one that is not interacting with any other 
part of the universe) remains constant regardless of any changes 
taking place within the system. For instance, the total quantity 
of energy within a closed system is always the same regardless 
of changes within the system and this is called "the law of con- 
servation of energy." 

The law of conservation of energy is a great convenience to 
physicists and is probably the most important single conservation 
law, and therefore the most important single law of any kind in 
all of science. Yet it does not seem to carry a note of overwhelm- 
ing necessity about it. 

Why should energy be conserved? Why shouldn't the energy of 
a closed system increase now and then, or decrease? 

Actually, we can't think of a reason, if we think of energy only. 
We simply have to accept the law as fitting observation. 

The conservation laws, however, seem to be connected with 
symmetries in the universe. It can be shown, for instance, that if 
one assumes time to be symmetrical, one must expect energy to 
be conserved. That time is symmetrical means that any portion of 


it is like any other and that the laws of nature therefore display 
"invariance with time" and are the same at any time. 

In a rough and ready way, this has always been assumed by 
man kin d — for closed systems. If a certain procedure lights a fire 
or smelts copper ore or raises bread dough on one day, the same 
procedure should also work the next day or the next year under 
similar conditions. If it doesn't, the assumption is that you no 
longer have a closed system. There may be interference from the 
outside in the form (mystics would say) of a malicious witch or 
an evil spirit, or in the form (rationalists would say) of unex- 
pected moisture in the wood, impurities in the ore, or coolness in 
the oven. 

If we avoid complications by considering the simplest possible 
for ms of matter — subatomic particles moving in response to the 
various fields produced by themselves and their neighbors — we 
readily assume that they will obey the same laws at any moment 
in time. If a system of subatomic particles were to be transferred 
by some time machine to a point in time a century ago or a 
million years ago, or a million years in the future, the change in 
time could not be detected by studying the behavior of the sub- 
atomic particles only. And if that is true, the law of conservation 
of energy is true. 

Of course, invariance with time is just as much an assumption 
as the conservation of energy is, and assumptions may not square 
with observation. Thus, some theoretical physicists have specu- 
lated that the gravitational interaction may be weakening in 
intensity very slowly with time. In that case, you could tell an 
abrupt change in time by noting (in theory) an abrupt change in 
the strength of the gravitational field produced by the particles 
being studied. Such a change in gravitational intensity with time 
has not yet been actually demonstrated, but if it existed, the law 
of conservation of energy would be not quite true. 

Putting that possibility to one side, we end with two equiv- 
alent assumptions: 1) energy is conserved in a closed system, 
and 2) the laws of nature are invariant with time. 

Either both statements are correct or both are incorrect, but it 


is the second, it seems to me, that seems more intuitively neces- 
sary to us. We might not be bothered by having a little energy 
created or destroyed now and then, but we would somehow feel 
very uncomfortable with a universe in which the laws of nature 
changed from day to day. 

Consider, next, the law of conservation of momentum. The total 
momentum (mass times velocity) of a closed system does not 
vary with changes within the system. It is the conservation of 
momentum that allows billiard sharps to work with mathematical 
precision. (There is also an independent law of conservation of 
angular momentum, where circular movement about some point 
or line is considered.) 

Both conservation laws, that of momentum and that of angular 
momentum, depend on the fact that the laws of nature are in- 
variant with position in space. In other words, if a group of sub- 
atomic particles is instantaneously shifted from here to the 
neighborhood of Mars, or of a distant galaxy, you could not tell 
by observing the subatomic particles alone that such a shift had 
taken place. (Actually, the gravitational intensity due to neighbor- 
ing masses of matter would very likely be different, but we are 
dealing with the ideal situation of fields originating only with the 
particles within the closed system, so we ignore outside gravita- 

Again, the necessity of invariance with space is more easily ac- 
cepted than the necessity of the conservation of momentum or of 
angular momentum. 

Most other conservation laws also involve invariances of this 
sort, but not of anything that can be reduced to such easily in- 
tuitive concepts as the symmetry of space and time. — Parity is an 

In 1927, the Hungarian physicist Eugene P. Wigner showed that 
conservation of parity is equivalent to right-left symmetry. 

This means that for parity to be conserved there must be no 
reason to prefer the right direction to the left or vice versa in 


considering the laws of nature. If one billiard ball hits another to 
the right of center and bounces off to the right, it will bounce off 
to the left in just the same way if it hits the other ball to the left 
of center. 

If a ball bouncing off to the right is reflected in a mirror that is 
held parallel to the original line of travel, the moving ball in the 
mirror seems to bounce off to the left. If you were shown diagrams 
of the movement of the real ball and of the movement of the 
nrirror-image ball, you could not tell from the diagrams alone, 
which was real and which the image. Both would be following 
the laws of nature perfectly well. 

If a billiard ball is itself perfectly spherical and unmarked it 
would show left-right symmetry. That is, its image would also be 
perfectly spherical and unmarked, and if you were shown a photo- 
graph of both the ball itself and the image, you couldn't tell 
which was which from the appearance alone. Of course, if the 
billiard ball had some asymmetric marking on it, like the number 
7, you could tell which was real and which was the image, because 
the number 7 would be "backward" on the image. 

The trickiness of the nrirror-image business is confused because 
we ourselves are asymmetric. Not only are certain inner organs 
(the liver, stomach, spleen, and pancreas) to one side or the other 
of the central plane, but some perfectly visible parts (the part 
in the hair, as an example, or certain skin markings) are also. 
This means we can easily tell whether a picture of ourselves (or 
some other familiar individual) is of us as we are or of a mirror 
image by noting that the part in the hair is on the "wrong side," 
for instance. 

This gives us the illusion that telling left from right is an easy 
thing, when actually it isn't. Suppose you had to identify left and 
right to some stranger where the human body could not be used 
as reference, to a Martian who couldn't see you, for instance. You 
might do it by reference to the Earth itself, if the Martian could 
make out its surface, for the continental configurations are asym- 
metric, but what if you were talking with someone far out near 
Alpha Centauri. 


The situation is more straightforward if we consider subatomic 
particles and assume them (barring information to the contrary) 
to be left-right symmetric, like perfectly spherical unmarked bil- 
liard balls. In that case if you took a photograph of the particle 
and of its mirror image, you could not tell from the appearance 
alone which was particle and which mirror image. 

If the particle were doing something toward our left, then the 
mirror image would be doing the equivalent toward our right. If, 
however, both the leftward act and the rightward act were equally 
possible by the laws of nature, you still couldn't tell which was 
particle and which was mirror image. — And that is precisely the 
situation that prevails when the law of conservation of parity 
holds true. 

But what if the law of conservation of parity is not true under 
certain conditions. Under those conditions, then, the particle is 
asymmetric or is working asymmetrically; that is, doing something 
leftward which can't be done rightward, or vice versa. In that 
case, you can say, "This is the particle and this is the image. I can 
tell because the image is backward (or because the image is doing 
something which is impossible)." 

This is equivalent to recognizing that a representation of a 
friend of ours is actually a mirror image because his hair part is 
on the wrong side or because he seems to be writing fluently with 
his left hand when you know he is actually right-handed. 

When Lee and Yang (see Chapter 1) suggested that the law 
of conservation of parity didn't hold in weak nuclear interactions, 
that meant one ought to be able to differentiate between a weak 
nuclear event and its mirror image. — And one common weak 
nuclear event is the emission of an electron by an atomic nucleus. 

The atomic nucleus can be considered as a spinning particle, 
which is symmetrical east and west and also north and south (just 
as the Earth is). If we take the mirror image of the particle (the 
"image-particle"), it seems to be spinning in the "wrong direc- 
tion," but are you sure? If you turn the image-particle upside 
down, it is then spinning in the right direction and it still looks 



just like the particle. You can't differentiate between the particle 
and the image-particle by the direction of its spin because you 
can't tell whether the particle or the image-particle is right side up 
or upside down. As far as spin is concerned, an upside-down 
image-particle looks just like a right-side-up particle. 

Of course, a spinning particle has two poles, a north pole and 
a south pole, and to all appearances we can tell which is which. By 
lining the particle up with a strong magnetic field we can com- 
pare the direction of the particle's axis of rotation with that of the 
Earth and identify the north and south pole. In that way we 
could tell whether the particle was right side up or upside down. 

Ah, but we are using the Earth as a reference here and the 
Earth is asymmetric thanks to the position and shape of the con- 
tinents. If we didn't use the Earth as reference (and we shouldn't 
because we ought to be able to work out the behavior of sub- 
atomic particles in deep space far from the Earth) there would 
be no way of telling north pole from south pole. Whether we con- 
sidered spin or poles, we couldn't tell a symmetrical particle from 
its mirror image. 

But suppose the particle gives off an electron. Such an electron 
tends to fly off from one of the poles, but from which? Suppose 
it could fly off from either pole with equal ease. In that case, if we 
were dealing with a trillion nuclei giving off a trillion electrons, 
half would fly off one pole and half off the other. We could not 
distinguish one pole from the other and we still couldn't distin- 
guish the particle from the image-particle. 

On the other hand, if the electrons tended to come off from one 
pole more often than from the other, we would have a marker for 
one of the poles. We could say, "Viewing the particle from a point 
above the pole that gives off the electrons, it rotates counter- 
clockwise. That means that this other particle is actually an image- 
particle, because viewed in that manner it rotates clockwise." 

This is exactly what should be true if the law of conservation 
of parity does not hold in the case of electron emission by nuclei. 

But is it true? When atomic nuclei (trillions of them) are 
shooting off electrons, the electrons come off in every direction 


equally — but that is only because the nuclear poles are facing in 
every direction, in which case electrons would shoot off in all ways 
alike whether they were coming from one pole only or from both 
poles equally. 

In order to check whether the electrons are coming from both 
poles or from one pole only, the nuclei must be lined up so that 
all the north poles are pointing in the same direction. To do this, 
the nuclei must be lined up by a powerful magnetic field and must 
be cooled to nearly absolute zero so that they have no energy that 
will vibrate them out of line. 

After Lee and Yang made their suggestion, Madame Chien- 
Shiung Wu, a fellow physics professor at Columbia University, 
performed exactly this experiment. Cobalt-6o nuclei, lined up ap- 
propriately, shot electrons off the south pole, not the north pole. 

In this way, it was proven that the law of conservation of parity 
did not hold for weak nuclear interactions. This meant one could 
distinguish between left and right in such cases, and the electron, 
when involved in weak nuclear interactions, tended to act left- 
ward rather than rightward, so that it can be said to be "left- 

The electron, which carries a unit negative electric charge, has 
another kind of "image." There is a particle exactly like the elec- 
tron, but with a unit positive electric charge. It is the "positron." 

Indeed every charged particle has a twin with an opposite 
charge, an "antiparticle." There is a mathematical operation which 
converts the expression that describes a particle into one that de- 
scribes the equivalent antiparticle (or vice versa). This operation 
is called "charge conjugation." 

As it happens, if a particle is left-handed, its antiparticle is 
right-handed, and vice versa. 

Observe then, that if an electron is doing something left- 
handedly, its mirror image would seem to be an electron doing it 
right-handedly, which is impossible — and the impossibility would 
serve to distinguish the image from the particle. 

On the other hand, if you employed the charge conjugation 


operation, you would change a left-handed electron into a left- 
handed positron. The latter is also impossible and this impos- 
sibility would serve to distinguish the image fiom the particle. 

In weak nuclear interactions, then, not only does the law of 
conservation of parity break down, but also the law of conserva- 
tion of charge conjugation.* 

However, suppose you not only alter the right-left of the elec- 
tron by imagining its mirror image, but also imagine that at the 
same time you have altered the charge from negative to positive. 
You have effected both a parity change and a charge conjugation 
change. The result of this double shift would be the conversion of 
a left-handed electron into a right-handed positron. Since left- 
handed electrons and right-handed positrons are both possible, 
you cannot tell by simply looking at a diagram of each, which is 
the original particle and which the image. 

In other words, although neither parity nor charge conjugation 
is conserved in weak nuclear interactions, the combination of the 
two is conserved. Using abbreviations we say that there is neither 
P conservation nor C conservation in weak nuclear interactions, 
but there is, however, CP conservation. 

It may not be clear to you how it is possible for two items to be 
individually not conserved, yet to be conserved together. Or (to 
put it in equivalent fashion) you may not see how two objects, 
each easily distinguishable from its mirror image, are no longer so 
distinguishable if taken together. 

Well, then, consider — 

The letter b, reflected in the m ir ror is d. The letter d, reflected 
in the minor is b. Thus, both b and d are easily distinguished 
from their mirror images. 

On the other hand, if the combination bd is reflected in a 
mirror, the image is also bd. Both b and d are individually in- 
verted and the order in which they occur is inverted, too. All the 

* Both conservation laws are true in strong nuclear interactions, however. 
In strong nuclear interactions, not only are leftward and rightward equally 
natural at all times, but anything a charged particle can do. the oppositely 
charged antiparticle can also do. 


inversions cancel and the net result is that although b and d are 
altered by reflection, the combination bd is not. (Try it yourself 
with printed lower-case letters and a mirror.) 

Let's point out one more thing about left-right reflection. Sup- 
pose the solar system were reflected in a m ir ror. If we observed 
the image, we would see that all the planets were circling the Sun 
the "wrong way" and that the Moon was circling the Barth the 
"wrong way," and that the Sun and all the planets were rotating on 
their axes the "wrong way." 

If you ignored the asymmetry of the surface structure of the 
planets, and just considered each world in the solar system to be 
a featureless sphere, then you could not tell the image from the 
real thing from their motions alone. The fact that everything was 
turning the "wrong way" means nothing, for if you observe the 
image while standing on your - head, then everything is turning 
the "right way" again, and in outer space there is no way of dis- 
tinguishing between standing "upright" and standing "on your 

And certainly the gravitational interaction, which is the pre- 
dominant factor in the solar' system's working, is unaffected by 
the reversal of right and left. If all the revolutions and rotations 
in the solar' system were suddenly reversed, gravitational interac- 
tions would account for the reversed motions as adequately and as 
neatly as for the originals. 

But consider this — 

Suppose that we didn't use a m ir ror at all. Imagine, instead, 
that the direction of time reversed itself. The result would be like 
that of running a movie film backward. With time reversed, the 
Earth would seem to be going "backward" about the Sun. All the 
planets would seem to be going "backward" about the Sun, and 
the Moon to be going "backward" about the Earth. All the bodies 
of the solar system would be spinning "backward" about their axis. 

But notice that the "backward" that takes place on reversing 
time, is just the same as the "wrong way" that takes place in the 



mirror image. Reversing the direction of time flow and mirror- 
imaging space produce the same effect. And there is no way of 
telling from observing the motions of the solar system alone 
whether time is flowing forward or backward. This inability to 
tell the direction of time flow is also true in the case of sub- 
atomic reactions (T conservation).* 

Or consider this — 

An electron moving through a magnetic field pointing in a par- 
ticular direction will veer to the right. The positron, with an op- 
posite charge, would, when moving in the same direction through 
the same magnetic field, veer to the left. The two motions are 
mirror images, so that in this case the shift from a charge to its 
opposite also produces the same effect as a left-right shift. 

Or suppose we reverse the direction of time flow. An electron 
moving through a magnetic field may veer to its right, but if a 
picture is taken of the motion and the film is reversed and pro- 
jected, the electron will seem to be moving backward and, in 
doing so, will veer to its left. Again, time flow and left-right sym- 
metry are connected. 

It would seem then that charge conjugation (C), parity (P), 
and time reversal (T) are all rather closely related and all some- 
how connected with left-right symmetry. If, then, left-right sym- 
metry breaks down in weak nuclear interaction with respect to 
one of these, the symmetry can be restored with one or both 

If a particle is doing something leftward, and its image is doing 
something rightward, which is impossible (so that the image can 
be spotted through a breakdown in P conservation), you can re- 
verse the charge on the image-particle and convert the action into 
a possibility. If the action is impossible even with the reversed 
charge (so that the image can be spotted through a breakdown 
in CP conservation), you can reverse the direction of time flow, 

* We can tell the direction of time flow under ordinary conditions easily 
enough because of entropy-change effects. This produces the equivalent of 
an asymmetry in time. Where entropy change is zero, however, as in plane- 
tary motions and subatomic events, T is conseived. 


and then you will find the action is possible. In other words, there 
is "CPT conservation" in the weak nuclear interaction.* 

The result is that the universe is symmetrical, as it has always 
been thought to be, with respect to strong nuclear interactions, 
electromagnetic interactions and gravitational interactions. 

Only weak nuclear interactions have been in question and there 
the failure of the law of conservation of parity seemed to intro- 
duce a basic asymmetry to the universe. The broadening of the 
concept to CPT conservation restored the symmetry — but only 
in theory. 

Does CPT conservation actually present us with a symmetrical 
universe in practice? As far as P (parity) is concerned, there is an 
equal supply of Tightness and leftness in the universe. As far as 
T (time reversal) is concerned, there is also an equal supply of 
pastness and futureness. But where C (charge conjugation) is 
concerned, symmetry in practice breaks down. 

The most common subatomic particles to be involved in weak 
nuclear interactions are the electron and the neutrino. For sym- 
metry to exist in practice, then, there should be equal supplies of 
electrons and positrons and equal supplies of neutrinos and anti- 
neutrinos. This, however, is not so. 

Certainly on Earth, almost certainly throughout our Galaxy, 
and, for anything we know to the contrary, throughout the 
entire universe, there are vast numbers of electrons and neutrinos, 
and hardly any positrons and antineutrinos. 

The universe then — at least our universe — or at the very least 
our section of our universe — is electronically left-handed and that 
may have had an interesting effect on the development of life. 

In order to explain that, I must change the subject radically, 
however, and make a new start. That I will do in the next chapter. 

* Actually, there was some indication in recent years that CPT is not 
invariably conserved in weak nuclear interactions and physicists have been 
examining the possible consequences in rather perturbed fashion. However, 
all the returns don't seem to be in Here- and we'll have to wait and see. 


I currently do my writing in a two-room suite in a hotel, and 
about a month ago I became aware of someone banging loudly 
against the wall in the corridor outside. Naturally, I was furious. 
Did whoever it was not realize that within my rooms the most 
delicate work of artistic creation was going on? 

I stepped into the corridor and there, on a ladder, at the ele- 
vators, was an honest workingman banging a hole into the wall 
for some arcane purpose of his own. 

"Sir," I said, with frowning courtesy, "how long do you intend 
to make the world hideous with your banging at that hole?" 

And the horny-handed son of toil turned his sweat-streaked face 
in my direction and answered jauntily, "How long did it take 
Michelangelo to do the ceiling in the Sistine Chapel?" 

What could I do? I burst out laughing, went back to my cell, 
and worked cheerfully along to the tune of banging which I no 
longer resented since it was produced by an artist who knew his 
own worth. 

Things take as long as they take, in other words. And even 
Michelangelo's long stint on his back, painting that fresco, pales 
into insignificance in comparison to the length of the intervals it 
took to build some corner or other of the majestic structure of 

In the seventeenth century, for instance, a question arose about 
light which wasn't answered for 148 years, despite the fact that, 
till it was answered, no theory as to the nature of light could pos- 
sibly hold water. 

The story begins with Isaac Newton, who, in 1666, passed a 
beam of sunlight through a prism and found that the beam was 



spread out into a rainhow lik e band which he called a "spectrum." 
' • Newton felt that since light traveled in a straight path, it must 
be made up of a stream of veiy fine particles, moving at an enor- 
mous speed. These particles differed among themselves in some 
way so that they produced the sensations of different cold's. In 
sunlight, all the different particles were mixed evenly and the 
effect was to impress our eye as white light. 

In passing obhquely into glass, however, the light particles bent 
sharply in their path; that is, they were "refracted." Particles dif- 
fering in their color nature were refracted by different amounts 
so that the cold's in white light were separated within the glass. In 
an ordinary sheet of glass, with two parallel faces, the effect was 
reversed when the light emerged once more from the other side, 
so that the cold's were again merged into white light. 

In a prism, it was different. The light particles bent sharply 
when they entered one side of the triangular' piece of glass, and 
then bent a second time in the same direction on emerging 
through a second, non-parallel side. The colors, separated on en- 
tering the prism, were even farther separated on emerging. 

All this made excellent sense, and Newton backed it up with 
careful experimentation and reasoning. And yet exactly what was 
different about the particles that gave rise to the various colors, 
Newton couldn't say. 

His contemporary the Dutch physicist Christiaan Huygens sug- 
gested in 1678 that light was a wave phenomenon. This made it 
possible to explain the different color's easily. A light wave would 
have to have some particular' length, and light of different wave- 
lengths might well impress the eyes as of different color's (just as 
sound of different wavelengths impresses the ears as of different 

Still, waves had their own problems. All man's experience with 
waves (water waves, for instance, and sound waves) made it clear' 
that waves curved around obstructions. Light on the other hand 
traveled in a straight line past obstructions and cast sharp 

Huygens tried to explain that away by presenting a mathemati- 



cal line of reasoning that showed that the ability to curve about 
an obstruction depended on the length of the wave. If light waves 
were much shorter than sound waves or water waves, they would 
then not bend, detectably, about ordinary obstructions. 

Newton recognized the convenience of the wave theory, but 
could not go along with the suggestion of waves so tiny they 
would cast sharp shadows. He stuck to particles and such was his 
eventual prestige that scientists, by and large, went along with the 
particle theory of light in order not to place themselves in dis- 
agreement with Newton. 

But in 1669, a Danish physician, Erasmus Bartholinus — a 
thoroughly obscure individual — made an observation which as- 
sured him a place in the history of science, for it raised a question 
the giants could not answer. 

Bartholinus had received a transparent crystal which had been 
obtained in Iceland, so that it became known as "Iceland spar," 
where "spar" is an old-fashioned term for a non-metallic mineral.* 

The crystal was shaped like a rhombohedron (a kind of slanted 
cube), with six flat faces, each one parallel to the one on the op- 
posite side. Bartholinus was studying the properties of this crystal 
and I presume he placed it on a piece of paper with writing or 
printing on it, on one occasion. When he picked it up, he noticed 
that the writing or printing was double when viewed through the 

In fact, when one looked through the crystal, it turned out one 
was seeing double. Apparently each beam of light entering the 
crystal was refracted, but not all to the same extent. Part of the 
light was refracted a certain amount and the remainder another 
and greater amount, so that though one beam entered the crystal, 
two beams emerged. The phenomenon was called "double re- 

Any theory of light had to explain double refraction, and 
neither Huygens nor Newton could do so. Apparently, the waves, 
or particles, of light must fall into two sharply defined classes so 

* Actually, Iceland spar is a transparent variety of calcium carbonate, if 
that helps any. 



that one class can behave in one way and the other class in an- 
other. The two-way difference can have nothing to do with color, 
for all colors of light were equally double-refracted by Iceland 

Huygens' view of light waves was that they were "longitudinal 
waves"; that is, similar to sound waves in structure (though much 
shorter in length) and that they represented a series of compres- 
sions and rarefactions in the ether they passed through. Huygens 
did not see how such longitudinal waves could fall into two 
sharply different classes. 

Nor could Newton see how light particles could be divided into 
two sharp classes. He speculated, rather vaguely, that the particles 
might differ among themselves in some fashion analogous to the 
two opposed poles of a magnet, but he didn't follow that up, since 
he was at a loss for any way of finding evidence for the suggestion. 

Physicists were forced to back away. Bartholinus' observation 
didn't fit either of the current theories of light, so, as far as pos- 
sible, it was to be ignored. 

This was not wickedness on the part of scientists; nor the ob- 
tuse workings of a conspiratorial "establishment." On the con- 
trary, it makes sense. 

Suppose a piece doesn't seem to fit a jigsaw puzzle. If you stop 
everything and start worrying exclusively about that troublesome 
piece, you may never get anywhere. If, however, you ignore the 
piece and continue working at other parts of the jigsaw, using 
whatever system seems convenient, you may eventually reach a 
point where, through the other work, new understandings are 
reached, and suddenly the old piece that was once so troublesome 
fits into place with no trouble at all. 

Double refraction was not forgotten altogether, of course. Even 
as late as 1808, it was still sticking in the scientific gizzard, and 
the Paris Academy offered a prize for the best mathematical treat- 
ment of the subject. A twenty-three-year-old French army engi- 
neer, named Etienne Louis Malus (who accepted Newton's 
particle theory) decided to see what he could do in that direction. 


He got some doubly refracting crystals and began to experiment 
with them. As it happened, he did not win the prize, but he made 
an interesting observation and coined a phrase that entered the 
scientific vocabulary. 

From his room he could see out on the Luxembourg Palace 
and, at one time, sunlight was reflected from a window of that 
palace into his room. Idly, Malus pointed a doubly refracting crys- 
tal in that direction, expecting to look through it and see two 
windows. He did not! He saw only one window. 

Apparently what happened was that the window, in reflecting 
the sunlight, reflected only one of the two classes of light particles. 

Malus remembered that Newton had said that the light 
particle varieties might be analogous to the opposing poles of a 
magnet. Thinking along those lines, he felt that only one pole of 
light had been reflected, and that the beam shining into his room 
contained only particles with that one pole. 

Malus therefore spoke of the light beam that entered his room 
as consisting of "polarized light." The phrase stands to this day, 
even though it is based on a false speculation, and even though 
the notion of poles of light was, in actual fact, being killed dead 
even before Malus had made his observation. 

In 1801, you see, an English physician, Thomas Young, began 
a series of experiments in which he showed that one beam of light 
could somehow cancel another intermittently, so that the two 
would not combine to give a smooth field of light, but rather a 
series of bands, alternately light and dark. 

If light consisted of particles, such "interference" was extremely 
difficult to explain. How could one particle cancel another? 

If light consisted of waves, however, interference was childishly 
easy to explain. If light consisted of alternate rarefactions and 
compressions, for instance, then if two light beams fitted together 
so that the compressed area in one beam fell on the rarefied area 
in the other and vice versa, the two lights would indeed cancel out 
into darkness. 

Young was able to explain every characteristic of his interfer- 
ence pattern by Huygens' wave theory. To be sure, many physi- 



cists (especially English physicists) tried to object, in the name of 
Newton. However, not even the most glorious name can long 
resist observations that anyone can confirm and explanations that 
explain perfectly. — So the wave theory won out. 

Yet Young could not explain double refraction any better than 
Huygens had. 

But then, in 1817, a French physicist, Augustin Jean Fresnel, 
suggested that perhaps the light waves were not longitudinal 
after the fashion of sound waves, and did not represent alter- 
nate compressions and rarefactions in the ether. Perhaps, instead, 
they were "transverse waves," like those on water surfaces; waves 
which moved up and down at right angles to the line of propaga- 
tion of the wave. 

Transverse waves could explain interference just as well as longi- 
tudinal waves did. If two light beams merged, and one was waving 
up where the other was waving down, and vice versa, the two 
would cancel, and two lights would make darkness. 

Water waves, which serve as a model for light waves, can only 
move up and down at right angles to the two-dimensional water 
surface. A ray of light, however, has greater freedom. Imagine 
such a ray moving toward you. It could wave up and down, or 
right and left, or anything in between and always be waving at 
right angles to the direction in which it was moving. (You can see 
what this means concretely, if you tie one end of a long rope to a 
post and make waves in it, up and down, right and left, or 

Once such transverse waves were proposed, they were accepted 
with remarkably little trouble, for through them, the phenome- 
non of double refraction could finally be explained, 148 years 
after the problem had arisen. 

To see that, consider that the light waves in an ordinary beam 
of light could be waving in all possible directions at right angles to 
the direction of travel — up and down, left and right, and all de- 
grees of in-betweenness. That would represent ordinary or "un- 
polarized" light 



Suppose, though, there were some way of dividing the light 
into two varieties, one in which all the waves move up and down, 
and the other in which all the waves move left and right. 

For each wave in unpolarized light which vibrates obliquely, 
there would be a division into two waves, of lesser energy, of 
the permitted classes. 

If a particular wave were just at forty-five degrees to the vertical, 
just halfway between the up and down and the left and right, it 
would be divided into two waves, one up and down and the other 
left and right, each with half the energy of the original. If the 
oblique wave were nearer horizontal than vertical, then it would 
be broken up into two waves, with the left and right having the 
greater supply of energy. If it were nearer the vertical, then the up 
and down would end with the greater supply of energy. 

It is easy to show, in fact, that a beam of unpolarized light can 
be divided into two beams of equal energy, in one of which all 
the transverse waves are in one direction, while in the other all 
the transverse waves are in a plane at right angles to the first. 
Since in each case all the waves move in a single plane, the un- 
polarized beam of light can be viewed as broken up into two 
mutually perpendicular "plane -polarized" beams. 

But what causes light to break up into plane -polarized beams? 
Certain crystals do. Crystals are made up of serried ranks and 
files of atoms arranged in very orderly array. Light, in passing 
through, is sometimes compelled to take up waves in certain planes 

(You can see a crude analogy of this if you pass a rope through 
a picket fence and tie it to a pole somewhere on the other side. 
If you make up-and-down waves in the rope, they will pass 
through the opening between the pickets, so that the rope on the 
other side of the fence also waves. If you make waves left and 
right, the pickets on either side of the opening stop those waves 
and the rope on the other side of the pickets does not wave. If you 
make the rope wave in every which way, only those waves which 
will fit between the pickets at least partly will get through, and 
on the other side of the fence, whatever you do, there will only 



be up-and-down waves. The picket fence polarizes the "rope 

Crystals such as Iceland spar will permit only two planes of 
vibration, one perpendicular to the other. Unpolarized light en- 
tering Iceland spar' breaks up into two mutually perpendicular 
plane-polarized beams within. The two beams of polarized light 
interact differently with the atoms, travel at different velocities 
and the slower beam is refracted through a greater angle. The 
two bea ms take separate paths within the crystal and emerge in 
different places. It is for that reason that looking through Iceland 
spar causes you to see double, and Bartholinus' puzzle is solved. 

Plane polarization can also take place on reflection. If an 
unpolarized beam strikes a reflecting surface at an angle, it often 
happens that those particular' waves which occupy a certain plane 
are more efficiently reflected than those in other planes. The re- 
flected beam is then heavily or even entirely plane-polarized and 
Malus' puzzle is solved. 

In 1828, a Scottish physicist, William Nicol, introduced a new 
refinement to Iceland spar'. He sawed a crystal in half in a certain 
fashion* and cemented the halves together with Canadian balsam. 
When light enters the crystal, it splits up into two plane-polarized 
beams, which travel in slightly different directions and hit the 
Canadian balsam seam at slightly different angles. The one that 
hits it at the lesser angle to the perpendicular' passes through into 
the other half of the crystal and eventually emerges into open 
ah'. The one that hits it at the greater angle is reflected and never 
enters the other half of the crystal. 

In other words, a beam of unpolarized light enters the "Nicol 
prism" at one end and a single beam of plane-polarized light 
emerges at the other end. 

Now imagine two Nicol prisms lined up in such a way that a 

* I am tempted every once in a while to present diagrams, and on rare 
occasions I do. I am, however, primarily a word-man and I try not to lean 
on pictorial crutches. In this case, the exact manner of dividing the crystal 
doesn't affect the argument, so the heck with it. 


beam of light passing through one will continue on into the sec- 
ond. If the two Nicol prisms are lined up parallel, that is, with 
the atom arrangements oriented in identical fashion in both, the 
beam of polarized light emerging from the first passes also through 
the second without trouble. 

(It is like a rope passing through two picket fences in both of 
which the pickets are up and down. An up-and-down rope wave 
that passes between the pickets in the first fence will also pass 
between the pickets in the second.) 

But what if the two Nicol prisms are oriented perpendicularly 
to each other? The plane-polarized beam emerging from the first 
Nicol prism is refracted through a greater angle by the second one 
and is reflected from the Canadian balsam seam in it. No light at 
all emerges from the second prism. (If we go back to the picket 
fence analogy, and have the pickets in the second fence arranged 
horizontally, you will see that any up-and-down waves that get 
through the first fence will be stopped by the second. No rope 
waves of any kind can go through two fences in one of which the 
pickets are vertical and in the other horizontal.) 

Suppose, next, that you arrange to have the first Nicol prism 
fixed in place, but allow the second Nicol prism to be rotated 
freely. Arrange also an eyepiece through which you can look and 
see the light that passes through both Nicol prisms. 

Begin with the two Nicol prisms arranged in parallel fashion. 
You will see a bright light in the eyepiece. Slowly rotate the second 
prism, which is nearer your eye. Less and less of the light emerg- 
ing from the first prism can get through the second, since more 
and more of it is reflected at the second's Canadian balsam seam. 
The light you see becomes dimmer and dimmer as you rotate the 
second prism, until, when you have turned through ninety de- 
grees, you see no light at ah. The same thing happens whether you 
rotate the prism clockwise or counterclockwise. 

Using such a pan of Nicol prisms you can determine the plane 
of vibration of a beam of polarized light. Suppose such a beam 
emerges from the fixed Nicol prism, but you are not sure as to 
exactly how that prism is oriented. That means you don't know 



the location of the plane of vibration of the light emerging. In 
that case, you need only turn the rotating Nicol prism until the 
beam of light you see through it is at its brightest.* At that point, 
the second prism is oriented parallel to the first and from the 
position of the second you know the plane of vibration of the 
polarized light. 

For this reason the first, fixed, Nicol prism is called the "polar- 
izer," and the second, rotating, one, the "analyzer." 

Now imagine an instrument in which there is a space between 
polarizer and analyzer into which a standard tube can be placed 
containing some liquid transparent to light. To make sure con- 
ditions are always the same, the temperature is kept at a fixed 
level, light of a single fixed wavelength is used, and so on. 

If the tube contains distilled water, nothing happens to the 
plane of polarized light emerging from the polarizer. The air, the 
glass, the water all may and do absorb a trifle of light, but the 
analyzer continues to mark the plane at the same point. If a salt 
solution is used in place of distilled water, the same thing is true. 

But place sugar solution in the tube, and something new hap- 
pens. The light you see through the analyzer is now greatly 
dimmed and this is not the result of absorption. Sugar solution 
doesn't absorb light significantly more than water itself does. 

Besides, if you rotate the analyzer, the light brightens again. 
You can eventually get it as bright as it was originally, provided 
you completely alter the orientation of the analyzer. What it 
amounts to is that the sugar solution has rotated the plane of 
polarized light. Anything which does this is said to display "opti- 
cal activity." The instrument used to detect optical activity and 
measure its extent is called a "polarimeter." 

A useful polarimeter was first devised in 1840 by the French 
physicist Jean Baptiste Biot. He had pioneered in the study of 

* It isn't so easy to tell when the light is brightest, but there is a device 
whereby the circle of light you see is divided into two half-circles and you 
turn the prism until the two half-circles are equally bright, something easy to 


optical activity long before he devised the polaiimeter (to make 
his work easier and more precise) and even before Nicol had first 
constructed his prism. 

As early as 1813, for instance, Biot reported certain observations 
that were eventually interpreted according to the new transverse- 
wave theory. It turned out that a quartz crystal, correctly cut, ro- 
tated the plane of polarized light passing through it. What's more, 
the thicker the piece of quartz, the greater the angle through which 
the plane was rotated. And still further, some pieces of quartz ro- 
tated the plane clockwise and some rotated it counterclockwise. 

The usual way of reporting the clockwise rotation was to say 
that the plane of polarization had been rotated to the right. Ac- 
tually, this is a careless and ambiguous way of reporting it. If the 
plane is viewed as straight up and down, then the upper end of 
it is indeed rotated to the right when it is twisted clockwise, but 
the lower end is rotated to the left. Vice versa, in the case of 
counterclockwise rotation. 

However, once a phrase enters the literature it is har'd to change 
no matter how poor, inappropriate, or downright wrong it is. 
(Look at the phrase "polarized light" itself, for instance.) Conse- 
quently, something that rotates the plane of polarized light 
clockwise, is said to be "dextrorotatory" ("right-rotating") and 
something that rotates it counterclockwise is "levorotatory" 

What Biot had shown was that there were two kinds of quartz 
crystals, dextrorotatory and levorotatory. Using initials, we can 
speak of d-quartz and Z-quartz. 

As it happens, quartz crystals are rather complicated in shape. 
In certain varieties of those crystals, just those varieties which 
show optical activity, it can be seen that there are certain small 
faces that occur on one side of the crystal, but not the other, in- 
troducing an asymmetry. What's more, there are two varieties of 
such crystals, one of which has the odd face on one side, the other 
of which has it on the other. 

The two asymmetric varieties of quartz crystals are minor 



images. There is no way in which you can twist one variety through 
three-dimensional space in order to make it look like the other, 
any more than you can twist a right shoe so as to make it fit a left 
foot. And one of these varieties is dextrorotatory, while its mirror 
image is levorotatory. 

It was quite convincing to suppose that an asymmetric crystal 
will rotate the plane of polarized light. The asymmetry of the crys- 
tal must be such that the light beam, traveling through, must be 
constantly exposed to an asymmetric force, one which pulls, so 
to speak, more strongly in one direction than the other. So the 
plane twists and keeps on twisting at a steady rate the greater the 
distance it must pass through such a crystal. What's more, if a 
crystal twists the plane of light in one direction, it is inevitable 
that, all else being equal, the mirror-image crystal will twist the 
plane in the opposite direction. 

You might even argue further that any substance which will 
crystallize in either of two mirror-image forms will be optically 
active. Furthermore, if two mirror-image crystals are taken of the 
same substance and of the same thickness, and if all the circum- 
stances are equal (such as temperature and wavelength of light), 
then the two crystals will show optical activity to precisely the 
same extent — one clockwise, the other counterclockwise. 

And, indeed, all evidence ever gathered shows all of this to be 
perfectly correct. 

But then, Biot went on to spoil the whole thing by discover- 
ing that certain liquids, such as turpentine, and certain solutions, 
such as camphor in alcohol and sugar in water, are also optically 

This presents a problem. Optica] activity is tied in firmly with 
asymmetry in all work on crystals, but where is the asymmetry in 
the liquid state. None that any chemist could see in 1840. 

Once again, then, the solution of one problem in science served 
to raise another. (And thank heaven for that, or where would 
there be any interest in science?) Having solved Bartholinus' prob- 
lem and Malus' problem by establishing the existence of transverse 



light waves, science found itself with Biot's problem — how a liquid 
which seemed to have no asymmetry about it could produce an 
effect that seemed to be logically produced only by asymmetry. 

Which brings us to Louis Pasteur's first great adventure in sci- 
ence — next chapter. 


In the days when I was actively teaching, full time, at a medical 
school, there was always the psychological difficulty of facing a 
sullen audience. The students had come to school to study medi- 
cine. They wanted white coats, a stethoscope, a tongue depressor, 
and a prescription pad. 

Instead, they found that for the first two years (at least, as it 
was in the days when I was actively teaching) they were subjected 
to the "basic sciences." That meant they had to listen to lectures 
veiy much in the style of those they had suffered through in col- 

Some of those basic sciences had, at least, a clear' connection 
with what they recognized as the doctor business, especially anat- 
omy, where they had all the fun of slicing up cadavers. Of all the 
basic sciences, though, the one that seemed least immediately "rel- 
evant," farthest removed from the game of doctor-and-patient, 
most abstract, most collegiate, and most saturated with despised 
Ph.D.'s as professors was biochemistry. — And, of course, it was 
biochemistry that I taught. 

I tried various means of counteracting the natural contempt of 
medical student for biochemistry. The device which worked best 
(or, at least, gave me most pleasure) was to launch into a spirited 
account of "the greatest single discovery in all the history of medi- 
cine" — that is, the germ theory of disease. I can get very dramatic 
when pushed, and I would build up the discovery and its conse- 
quences to the loftiest possible pinnacle. 

And then I would say, "But, of course, as you probably all take 
for granted, no mere physician could so fundamentally revolution- 
ize medicine. The discoverer was Louis Pasteur, Ph.D., a bio- 


Pasteur's first great discovery, however, had nothing to do with 
medicine, but was a matter of straight chemistry. It involved the 
matter of optically active substances, a subject I discussed in the 
previous chapter. To see how he contributed, let's start at the be- 

In the wine-making process of the fermentation of grape juice, 
a sludgy substance separates and is called "tartar," a word of un- 
known origin. From this substance, the Swedish chemist Karl Wil- 
helm Scheele in 1769 isolated a compound which had acid 
properties and which he naturally called "tartaric acid." 

In itself this wasn't terribly important, but then in 1820, a Ger- 
man manufacturer of chemicals, Charles Kestner, prepared some- 
thing he felt ought to be tartaric acid and yet didn't seem to be. 
For one thing, it was distinctly less soluble than tartaric acid. A 
number of chemists obtained samples and studied it curiously. 
Eventually, the French chemist Joseph Louis Gay-Lussac named 
this substance "racemic acid" from the Latin word for a "cluster 
of grapes." 

The more closely racemic acid and tartaric acid were studied, 
the more puzzling were the differences in properties. Analysis 
showed that each acid had exactly the same proportion of exactly 
the same elements in their molecules. Using modern symbols, the 
formula for each compound was C |H f ,06. 

In the early nineteenth century, when the atomic theory had 
only been in existence for a quarter of a century or so, chemists 
had decided that every different molecule had a different atomic 
content, that it was, in fact, the difference of atomic content that 
was responsible for the difference of properties. Yet here were two 
substances, quite distinguishable, with molecules made up of the 
same proportions of the same elements. It was very disturbing, 
especially since this was not the first time such a thing had been 

In 1830, the staunchly conservative Swedish chemist Jons Jakob 
Berzelius,* who didn't believe that molecules with equal structures 

* I have a tendency (as you may occasionally have noticed) to mention 
large numbers of scientists and to give the contribution of each whenever 



but different properties were possible, examined both tartaric acid 
and racemic acid in detail. With considerable chagrin, he decided 
that even though he didn't believe it, it was nevertheless so. He 
bowed to the necessary, accepted the finding, and called such 
equal-structure-different-property compounds "isomers" from 
Greek words meaning "equal proportions" (of elements, that is). 

But how could isomers have the same atomic composition and 
yet be different substances? One possibility is that it is not just 
the number of atoms of each element that is distinctive, but their 
physical arrangement within the molecule. This thought, how- 
ever, was something chemists shuddered away from. The whole 
notion of atoms was a shaky one. Atoms were useful in explaining 
chemical properties but they could not be seen or detected in any 
way and they might very well be no more than convenient fictions. 
To start talking about actual arrangements within the molecules 
was to advance farther down the road of accepting atoms as real 
entities than most chemists cared to — or dared to. 

The phenomenon of isomerism was therefore left unaccounted 
for and kept suspended until such time as chemical advance might 
produce an explanation. 

One difference in properties between tartaric acid and racemic 
acid was particularly interesting. A solution of tartaric acid or of 
its salts (that is, compounds in which the acid hydrogen of the 
compound was replaced by an atom of such elements as sodium 
or potassium) was optically active. It rotated the plane of polar- 
ized light clockwise and was therefore dextrorotatory (see the 
previous chapter), so that the compound could well be called 
d-tartaric acid. 

A solution of racemic acid, on the other hand, was optically 
inactive. It did not rotate the plane of polarized light in either 
direction. This difference in properties was clearly demonstrated 

I get science-historical. This is not a matter of name-dropping. Every advance 
in science is the result of the co-operative labor of a number of people, and 
I like to demonstrate that. And I am careful to mention nationalities because 
it is also important to recognize the fact that science is international 
in scope. 



by the French chemist Jean Baptiste Biot, whom I mentioned in 
the previous chapter as a pioneer in the science of polarimetry. 

No one at the time knew why any substance should be optically 
active in solution, but they did know this — Those crystals known 
to be optically active had asymmetric structures. In that case, if 
one were to prepare crystals of tartaric acid and racemic acid or of 
their respective salts, it would surely turn out that those of the 
former were asymmetric and those of the latter, symmetric. 

In 1844, however, the German chemist Eilhardt Mitscherlich 
undertook this investigation. He formed crystals of the sodium 
ammonium salt of both tartaric acid and racemic acid, studied 
them carefully, and announced that the two substances had ab- 
solutely identical crystals. 

The basic findings of the budding science of polarimetry were 
blasted by this report and for the moment all was confusion. 

It was at this point that the young French chemist Louis Pas- 
teur entered the scene. He was only in his twenties and his scho- 
lastic record at school had been mediocre, yet he had the temerity 
to suspect it possible that Mitscherlich (a chemist of the first 
rank) might have been mistaken. After all, the crystals he studied 
were small and perhaps some tiny details were overlooked. 

Pasteur applied himself to the matter and began to produce the 
crystals and study them painstakingly under a hand lens. He finally 
decided that there was a definite asymmetry to the crystals of the 
sodium ammonium salt of tartaric acid. So far, so good. That, at 
least, was to be expected, since the substance was optically active. 

But was it possible now that the sodium ammonium salt of 
racemic acid yielded crystals of precisely the same sort, as 
Mitscherlich maintained? In that case, there would be asymmetric 
crystals of a substance which was not optically active, and that 
would be very unsettling. 

Pasteur produced and studied the crystals of the salt of racemic 
acid and found that they were indeed also asymmetric but that 
not all the crystals were identical. 

Some of the crystals were exactly like those of the sodium am- 



monium salt of tartaric acid, but others were mir ror images of the 
first group and were asymmetric in the opposite sense. 

Could it be that racemic acid was half tartaric acid and half 
the mirror image of tartaric acid, and that the reason racemic acid 
was optically inactive was that it was made up of two parts, one 
pail of which neutralized the effect of the other pail? 

This had to be checked directly. Making use of his hand crystal 
and a pair of tweezers, Pasteur began to work over those tiny 
crystals of the racemic acid salt. All those which were right-handed 
he shoved to one side; all those which were left-handed, to the 
other. It took him a long time, for he wanted to make no mistake, 
but he was eventually done. 

He then dissolved each set of crystals in a separate sample of 
water and found both solutions to be optically active! 

One of the solutions was dextrorotatory, exactly as tartaric acid 
was. In fact, it was tartaric acid, in eveiy sense. 

The other was levorotatory, and differed from tartaric acid in 
rotating the plane of polarized light in the opposite direction. It 
was Z-tartaric acid. 

Pasteur's conclusion, announced in 1848, when he was only 
twenty-six, was that racemic acid was optically inactive only be- 
cause it consisted of equal quantities of d-tartaric acid and Z-tartaric 

The announcement created a sensation and Biot, the grand 
old man of polarimetry, who was seventy-four year's old at the 
time, cautiously refused to accept Pasteur's finding. Pasteur there- 
fore undertook to demonstrate the matter to him in person. 

Biot gave the young man a sample of racemic acid which he had 
personally tested and which he knew to be optically inactive. Un- 
der Biot's shrewd, old eyes, alert for hanky-panky, Pasteur formed 
the salt, crystallized it, isolated the crystals, and separated them 
painstakingly by means of hand lens and tweezers. Biot then took 
over. He personally prepared the solutions from each set of crystals 
and placed them in the polarimeter. 

You guessed it. He found that both solutions were optically 


active, one in the opposite sense to the other. After that, with 
typical Gallic enthusiasm, he became fanatically pro-Pasteur. 

Actually, Pasteur had been most fortunate. When the sodium 
ammonium salt of racemic acid crystallizes, it doesn't have to form 
separate mirror-image crystals. It might also form combination 
crystals in each of which are equal numbers of molecules of 
d-tartaric acid and Z-tartaric acid. These combination crystals are 

Had Pasteur obtained these crystals he would still have noted 
their difference from those of the sodium ammonium salt of tar- 
taric acid and have refuted Mitscherlich. On the other hand, he 
would have missed the far greater discovery of the reason for the 
optical inactivity of racemic acid and he would also have missed 
having been the very first man to form optically active substances 
from an optically inactive start. 

As it happens, only symmetric-combination crystals are formed 
out of solutions above 28° C. (82° F.). It requires solutions of so- 
dium ammonium salt of racemic acid at temperatures below 28° 
C. to form separate sets of asymmetric crystals. Furthermore, the 
crystals formed are usually so tiny that they are far too small to 
separate with hand lens alone. It just happened that Pasteur was 
working at low temperatures and under conditions which pro- 
duced fairly good-sized crystals. 

Pasteur might be dismissed as an ordinary man who took ad- 
vantage of an unexpected good break, but (as I used to tell my 
biochemistry class) .he managed to take advantage of similarly 
unexpected good breaks every five years or so. After a while, you 
had to come to the conclusion that it was Pasteur who was re- 
markable and not the laws of chance. 

As Pasteur himself once said, "Chance favors the prepared 
mind." We all get our share of lucky breaks and the great man 
is he who is capable of recognizing a break when it comes, and of 
taking advantage of it. 

Pasteur continued to interest himself in the matter of the tar- 
taric acids. He found that if he heated d-tartaric acid for pro- 


longed periods under certain conditions, some of the molecules 
would change to the Z-fonn and racemic acid would be produced. 
(Ever since, the ability to change optical activity to optical in- 
activity by heat or by some chemical process through forma- 
tion of some of the oppositely active form has been known as 

Pasteur also found a kind of tartaric acid which was optically 
inactive, which could not be separated into opposite forms under 
any conditions, and which possessed properties distinct from those 
of racemic acid. He called it meso-tartaric acid, from the Greek 
word for "intermediate," since it seemed intermediate between 
the d- and the Z-fomis of the acid. 

But all these facts could not explain the existence of optical 
activity in solutions. Granted that some crystals are symmetrical, 
while others are asymmetric in one sense or the other, still there 
are no crystals in solution. There are only molecules. 

Could not the molecules themselves retain the asymmetry of 
the crystals? Was not the asymmetry of the crystals but a reflec- 
tion of that of the molecules that composed them? Was not 
racemization a result of the heat-induced rearrangement of atoms 
within the molecule? Pasteur was sure of all this, but he could 
think of no way of proving it or of demonstrating what the ar- 
rangements must be. 

In the 1860's, to be sure, the German chemist Friedrich August 
Kekule worked out a system whereby a molecule was pictured 
not merely as a conglomeration of so many atoms of this element 
or that, but as a collection of atoms connected to one another in a 
definite arrangement (see Chapter 13). Little dashes were used 
between symbols of the elements to represent the "bonds" link- 
ing one atom to another, so that the molecule did get to look like 
a Tinker Toy. 

However, the Kekule structures were considered to be highly 
schematic and to be merely another useful tool for chemists who 
were working out organic structures and reactions. As in the case 
of atoms themselves, chemists were not prepared to say that the 



Kekule structures actually represented the true situation within 
the molecules. 

The Kekule structures did explain the existence of many isomer's, 
since they demonstrated gross differences in atomic arrangement 
even when the total numbers of atoms of each element present 
within the molecule were the same. The Kekule structures did 
not, however (as used originally), account for those "optical 
isomers" which differed only in the way in which they twisted the 
plane of polarized light. 

We next come to the Dutch chemist Jacobus Hendricus van't 
Hoff, who took up the problem in 1874, when he was only twenty- 
two. The following represents what may have been his line of rea- 

According to the Kekule system, a carrion atom is represented 
by the letter C with four little bonds attached to it. Usually, these 
little bonds are shown pointing to the comers of an imaginary 

square, thus, j C I , so that the angle between any two adjacent 

bonds is ninety degrees. A carrion atom will combine with four 
hydrogen atoms to form the substance methane, which will then 
look like this: 

/ \ 

Are the four - bonds identical? If each is different from the rest, 
somehow, then what would happen if one of the hydrogen atoms 
is replaced by a chlorine atom to form "methyl chloride"? Surely, 
there would then be four - different methyl chlorides, depend- 
ing on which of the four different bonds the chlorine atom hap- 
pened to attach itself to. 

But there aren't. There is only one methyl chloride and no 
more. This indicates that the four' carrion bonds are equivalent 
and, indeed, if the four' are drawn to the comers of a square, that 



is what should be expected. One comer of the square should be 
no different from any other. 

Consider the situation, though, if two chlorine atoms replace 
hydrogen atoms to form "methylene chloride." Then, if we still 
deal with bonds pointing to the cornel's of a square, there ought 
to be two different methylene chlorides, depending on whether 
the two chlorine atoms are placed at adjacent comers of the square 
or at opposite comers, thus: 

H Cl H Cl 

V (R V 

h a ci h 

But there aren't. There is only one methylene chloride and no 
more, which shows that the Kekule structures can't possibly cor- 
respond to reality (and, of course, no one claimed that they did). 

One way in which they were almost certain not to correspond 
to reality was that all were drawn, for convenience' sake, in two 
dimensions — that is, in a plane — and surely it was unlikely that all 
molecules would be strictly planar in nature. 

The foui' bonds of the carbon atoms were almost certainly dis- 
tributed in three dimensions and it was only necessary to choose 
some 3-D arrangement in which each bond was equally adjacent 
to all three remaining bonds. Only then would there be only a 
single methylene chloride. 

The simplest way of arranging this was to have the four bonds 
pointing toward the apices of a tetrahedron.* The carbon atom 
then looks as though it were resting on three bonds forming a 
squat tripod while the fourth bond is pointing straight up. It 
doesn't matter which bond you point upward, the other three al- 
ways form the squat tripod. The carbon atom can thus stand in 
each of four different positions and look the same each time. 

* A tetrahedron is a solid bounded by four equilateral triangles. It can best 
be understood if it is inspected in the form of a three-dimensional model. 
Failing that, you are probably familiar with the shape of the Egyptian pyra- 
mids — a square base, with each wall slanting inward from one side of that 
base toward an apex on the top. Well, if you imagine a triangular base in- 
stead, you have a tetrahedron. 



What's more, any one bond is equally far from each of the other 
three. The angle between any two bonds is 109%°. 

If we deal with such a "tetrahedral carbon," then as long as two 
of the bonds are attached to identical atoms (or groups of atoms), 
it doesn't matter what atoms, or groups of atoms, are attached 
to the other two; in every case all possible arrangements are equiv- 
alent and only one molecule is formed. 

Thus, if attached to the four' bonds of a carbon atom are aaaa, 
or aaab, or aabb, or aabc, then it doesn't matter to which bond 
which atom is attached. If you attach them so as to form what 
seem to be two different arrangements, then by twisting the first 
arrangement so that some different bond faces upward, you can 
make it identical with the second. 

Not so when you have four different atoms or groups of atoms 
attached to the four bonds: abed. In that case, it turns out there 
are two different and distinct arrangements possible, one of which 
is the mir ror image of the other. No amount of twisting and turn- 
ing can then make one arrangement look like the other. 

A carbon atom to which four' different atoms or groups of atoms 
are attached is an "asymmetric carbon." 

It turns out that optically active organic substances invariably 
have asymmetric molecules if the Van't Hoff system is used. Al- 
most always there is at least one asymmetric carbon present. 
(Sometimes there is an asymmetric atom other than carbon pres- 
ent and sometimes the molecule as a whole is asymmetric even 
though none of the carbon atoms are.) 

In tartaric acid there are present two asymmetric carbon atoms. 
Either can be present in a certain configuration or in its mirror 
image. Let's refer to these arbitrarily as p and q (since q is the 
mir ror image of p ). If the two carbon atoms are pp, then we have 
d-tariaric acid and if qq, Z-tartaric acid. 

If the two halves of the molecule, each with one asymmetric 
carbon, were not identical, we would have two other optically ac- 
tive forms, pq and qp. In the case of tartaric acid, however, the 
two halves are identical in structure, so that pq and qp are identi- 
cal and, in each case, the optical activity of one half balances the 



optical activity of the other. The net result is optical inactivity, 
and we have meso-tartaric acid. 

It is not easy to see all this without careful structural formulas, 
which I will not plague you with. The crucial point to remember 
is that from 1874 right down to the present day, all questions of 
optical activity, no matter how involved, have been satisfactorily 
explained by a careful consideration of the tetrahedral carbon atom 
together with similar structures for other atoms. Although our 
knowledge of atomic structure has enormously expanded and 
grown vastly more subtle in the century since, Van't Hoffs ge- 
ometrical picture remains as useful as ever. 

Van't Hoffs paper dealing with the tetrahedral atom appeared 
in a Dutch journal in September 1874. Two months later, a some- 
what similar paper appeared in a French journal. The author was 
a French chemist, Joseph Achille Le Bel, who was twenty-seven 
at the time. 

The two young men worked it out independently, so that both 
are given equal credit and one usually speaks of the Van't Hoff-Le 
Bel theory. 

The tetrahedral atom did not at once meet with the approval 
of all chemists. After all, there was still no direct evidence that 
atoms existed at all (and nothing direct enough to be convincing 
was to come for another generation). To some of the older and 
more conservative chemists, therefore, the new view, placing atom 
bonds just so, smacked of mysticism. 

In 1877, the German chemist Hermann Kolbe, then fifty-nine 
years old and full of renown, published a strong criticism of Van't 
Hoff and his views. It was quite within Kolbe's right to criticize, 
for it could be argued that the new view went beyond the foun- 
dations of chemistry as they then existed. 

In fact, an essential part of the practical working of the sci- 
entific method is that new ideas be subjected to searching criti- 
cism. They must be jumped at and hammered down in fair and 
sporting fashion, for one of the tests of the value of the new idea 
is its ability to survive hard knocks. 



Kolbe, however, was neither fair nor sporting. He characterized 
Van't Hoff as a "practically unknown chemist," which had noth- 
ing to do with the case. Even more unforgivably, he sneered at 
him for holding a position at the Veterinary School of Utrecht, 
managing to refer to it three times in a short space, thus exhibit- 
ing a rather unlovely professorial snobbeiy. 

Nevertheless, to those who think that the scientific "establish- 
ment" has the power to quash useful advances permanently at the 
simple behest of conservatism and snobbeiy, let it be stated that 
the tetrahedral atom was adopted with reasonable speed. It 
worked so well that not all of Kolbe's sour fulminations could 
stop it and Van't HofFs career went on untouched. (In fact, Van't 
Hoff rapidly became one of the leading physical chemists in the 
world and in 1901, when the Nobel prizes were established, the 
first award in chemistry went to him.) 

Kolbe is today best known, perhaps, not for his own veiy real 
contributions to chemistry, but for his diatribe against Van't Hoff 
— which is reprinted to amuse the audience.* 

And again a new advance meant new problems. Once the struc- 
ture of the carbon atom and its bonds had been worked out, and 
the details of molecules described in 3-D, a curious asymmetry 
turned out to exist in living tissue. That will be the subject of the 
next chapter. 

* I was recently challenged to give my views on a book of far-out theory 
by someone who said he wanted my views especially if unfavorable, as he was 
making a collection which would someday, in hindsight, make very amusing 
reading. The book of far-out theory seemed like nonsense to me but I was 
aware of Kolbe's misfortune and I hesitated. But then I decided that I was 
not going to duck the issue out of fear for posterity's views. I thought the 
theories were worthless and I said so. However, I was polite about it. That 
much costs nothing. 


Only yesterday (as I write this) I was on a Dayton, Ohio, talk 
show, by telephone, one of those talk shows where the listeners 
are encouraged to call in questions. 

A young lady called in and said, "Dr. Asimov, who, in your 
opinion, did the most to improve modern science fiction?" 

I answered, after the barest hesitation, "John W. Campbell, Jr."* 

Whereupon she said, "Good! I'm Leslyn, his daughter." 

I carried on, of course, but inside I had a momentary dizzy spell. 
The reason for my second's hesitation in answering was that I 
had had to make a quick choice between two alternatives. I could 
have answered honestly and said, "Campbell!" as I did; or I could 
have played it for laughs, as I so often do, and said, "Me!" If I had 
had a visible audience and could have relied on hearing the laugh, 
I would undoubtedly have opted for the joke. As it was, with no 
possibility of a tangible reaction, I played it, thank goodness, 
straight — and avoided what would have been a terrible embarrass- 

Well, it sometimes happens, in science, that a person has a 
choice of two alternatives and has to face the possibility that his 
choice, whichever it is, will stamp itself indelibly on the field. If he 
guesses wrong, that wrongness may be impossible to remove and 
will be a source of endless posthumous embarrassment. 

Thus Benjamin Franklin once decided that there were two types 
of electric fluid and that one of them was mobile and one sta- 

* John Campbell, who died on July 11, 1971, was, in my opinion (and 
that of many others) the outstanding personality of all time in the field of 
science fiction. I owe a personal debt to him past all calculation. I have said 
this elsewhere. I wish to say so here. 


tionary. Thus some substances, when nibbed, gained an excess 
•(+) of the mobile fluid, while others lost some of the mobile 
fluid and suffered a deficit ( — ). The one with the deficit showed 
the effect of the excess of the other, stationary fluid, so we could 
say that the two substances, (+) and ( — ), would show opposite 
electrical effects. 

And so they do. An amber rod and a glass rod show opposite 
electrical effects when rubbed. (They attract each other, once 
charged, instead of repelling each other as like charges — two glass 
rods, for instance — would.) The question was: Which had the ex- 
cess of the movable fluid and which the deficit; which was (+) 
and which was ( — -)? 

There was absolutely no way of telling and Fran kli n was forced 
to guess. He guessed the amber had the excess, assigned it (+) 
and the glass he assigned ( — ). That set the standard. All other 
charges were traced back to Franklin's decision on amber vs. glass 
and to the present day it is usually assumed in electrical engineer- 
ing that the current flows from the positive terminal to the nega- 

By Franklin's standard the first two fundamental subatomic 
particles of ordinary matter were assigned their charge, too. The 
electron which tends to move toward the positive terminal is as- 
signed ( — ); and the proton which is attracted to the electron is 
(+). They represent, in a sense, Franklin's two electric fluids, 
but, as it happens, it is the electron that is mobile and the proton 
that is relatively stationary, so that the current really flows from 
the negative terminal to the positive. 

Fran kli n had had a fifty-fifty chance of guessing right, and he 
muffed it. Too bad. Fortunately, the wrong guess had no effect 
on the practical development of electrical technology or even on 
theory — but it always represents an irritating bit of non-neatness 
to neat-nuts like myself. 

In this chapter, however, we will, in passing, mention another 
fifty-fifty choice of alternatives and see how that worked out. 

Once again, we are dealing with optical isomerism, the subject 
of the previous two chapters. Van't Hoff and Le Bel had shown 


(as I explained in Chapter 4) that if the four bonds of a carbon 
atom were attached to four different kinds of atoms or groups of 
atoms, that carbon atom was "asymmetric." The four attached 
groups could be attached in either of two possible configurations 
which were essentially different, one being the mirror image 
of the other. 

A compound containing an asymmetric carbon atom can, in 
other words, be "left-handed" or "right-handed." 

As we might expect, nature has no left-right bias in this respect. 
Two compounds which differ, structurally, only in being left- 
handed or right-handed have identical chemical and physical prop- 
erties and, when faced with conditions which are not themselves 
asymmetric, always react in the same way. 

We might make an analogy to the right and left hand (or foot, 
or eye, or nostril, or upper canine). In each case the two organs 
have identical features and functions. What one can do the other 
can do and generally in equal fashion. The mirror imagery is not 
perfect, perhaps. The right and left hand of a given individual 
don't have mirror-image fingerprints, for instance. Also, most peo- 
ple use one hand with greater ease than the other — but that is 
because the brain itself is not perfectly symmetrical. 

Chemical compounds, which are less complicated than the hu- 
man hand, demonstrate left-right symmetry to a much higher de- 
gree of perfection than hands do. What a left-handed molecule 
can do, its right-handed brother can also do, and just as well. 

(Of course, an equal mixture of right-handed and left-handed 
twins may have some properties which differ from those of either 
separately, as in the case of racemic acid and tartaric acid described 
in the previous chapter, but that's a different matter. A right hand 
and left hand clasped together can be easily distinguished from 
two rights — or two lefts — clasped together, and because of the dif- 
fering position of the thumbs, undoubtedly function differently.) 

To see the significance of right-left symmetry, suppose you be- 
gin with a molecule that contains no asymmetric carbon and sub- 
ject it to a chemical change that produces one. Thus, if a carbon 
has attached to it abcc, and you change one of the attached c's 


to a d, so that the whole becomes abed, a symmetric carbon be- 
comes an asymmetric one. 

The d can replace either of the two c's. If it replaces one, there 
results a left-handed molecule and if it replaces the other, there 
results a right-handed molecule. The chances are exactly even; 
neither result is favored over the other. 

Consequently, in any reaction of this sort, almost exactly equal 
numbers of each twin are produced. Any deviation from exact 
equality (and some deviation is to be expected in any chance 
process) would not be large enough to be detectable. 

No matter what chemists do, short of introducing some asym- 
metric factor to begin with, they end up with symmetry. There 
seems no way of forcing Nature to make a right-left choice on 
the molecular level. 

You can work the other way round. You can have a mixture 
containing equal numbers of the left-handed and right-handed 
mirror-image molecules, and subject that mixture to some physical 
or chemical effect (that is not, itself, asymmetric) which will alter 
the molecules. The altered molecules are such that they can be 
easily separated from the original. If the effect, whatever it is, de- 
stroys the left-hand molecule a little more rapidly or easily than 
the right-hand molecule (or vice versa), what will be left after a 
time, will show an excess of one or the other. The mixture will 
end by being at least slightly asymmetrical. 

But that never happens either. You can't form molecular asym- 
metry out of a situation that is symmetrical to begin with. 

I have been careful to rule out asymmetric effects till now, but 
suppose we decide to use one — 

Suppose you have a substance made of two mirror-image twins 
in equal numbers; call them b and d, to use mirror-image letters. 
Next suppose you have another compound, which does not con- 
tain an asymmetric carbon atom, so that its molecules are sym- 
metric. Call it o, a symmetric letter. If o combines with b and d 
to form an addition compound, then bo and od will be formed. 
These are still mirror images and can't be separated. 


On the other hand, what happens if you have another com- 
pound which contains one or more asymmetric carbon atoms, so 
that it exists in right- and left-handed forms, and you actually have 
one or the other variety only? Call this p. 

Again you form an addition compound and end up with bp and 
pd, which are not mirror images. (The mirror image of bp is qd, 
not pd.) The addition compounds, not being mirror images, have 
different properties and can be easily separated. Once the addi- 
tion compounds are separated, each is broken down to b and p, 
or to p and d. The p is easily gotten rid of and the chemist is left 
with b and d in separate test tubes. He has two compounds, each 
of which is asymmetric and optically active, and this is called an 
"asymmetric synthesis." 

You might very well ask, though, where a chemist gets the 
asymmetric p in the first place? If he can end with an asymmetric 
compound only when he begins with one, isn't he working in a 
circle? Where does the first asymmetric compound come from? 

As it happens, it is easy to find compounds that are already 
asymmetric — but with an important restriction. He can find them 
only in connection with life. In fact, asymmetric compounds exist 
in nature only in living tissue or in matter that was once part of 
living tissue. 

In fact, we can go farther than that. There are numerous mole- 
cules that have one or more asymmetric carbon atoms and that 
are to be found in living tissue. In every case only one of the op- 
tically active pairs is to be found there. If a left-hand compound 
is found in living tissue, the right-hand mirror image is not; if a 
right-hand compound is found in living tissue, the left-hand image 
is not* 

What's more, the choice between one twin and the other does 
not vary from species to species. If the left-handed twin is favored 
in the living tissue of any one species, it is favored in all living 
tissue of all species. All of earthly life makes use of only a single 

* Actually, the non-occurring mirror images occasionally do occur, in spe- 
cialized places and in very limited amounts. Their very trifling presences 
merely emphasize the general rule. 



one of any molecule capable of existing as minor-image twins, 
and always the same single one. 

(This accounts, by the way, for the fact that Pasteur could sep- 
arate the mirror-image components of racemic acid mechanically, 
as described in the previous chapter. Pasteur, being alive, was him- 
self asymmetric.) 

Is there perhaps some regularity to be found in which mirror- 
image twins will occur in tissue. At first glance, it doesn't seem 
so. Some compounds in living tissue are dextrorotatory and some 
are levorotatory and there seems no regularity to the matter. For 
instance, consider two very common sugars in living tissue: "glu- 
cose" and "fructose." Both are made up of the same number of the 
same atoms and are very similar in properties. However, glucose is 
dextrorotatory and fructose, levorotatory, so that we have d- 
glucose and Z-fructose. 

Nor are these mirror images, I hasten to say. Each does have 
a mirror image, Z-glucose and d-fructose, respectively, which do not 
occur in living tissue. 

Once the Van't Hoff-Le Bel theory was advanced in 1874, some- 
thing more than mere optical rotation was possible as a way of 
characterizing the mirror-image twins. Why not determine the 
actual configuration of the various groups about the asymmetric 
carbon atom and see if any regularity among the compounds found 
in living tissue follows from that? 

This project was undertaken by the German chemist Emil 
Fischer, who began working with sugar molecules in the 1880's. 
A molecule such as that of glucose has six carbon atoms, of which 
no less than four are asymmetric. Each one of the four can exist 
as a pair of mirror images, so that there are altogether sixteen dif- 
ferent glucoselike compounds, arranged in eight pairs of minor 

To simplify matters, Fischer began with the simplest possible 
sugarlike compound, glyceraldehyde. It has three carbon atoms, 
of which only one is asymmetric. Glyceraldehyde therefore exists 



as just one pair of mirror-image twins, d-glyceraldehyde and l- 

The foui' different groups about the single asymmetric carbon 
atom in glyceraldehyde could be arranged in two different ways. 
Which arrangement should be assigned to the d-twin and which 
to the Z-twin? Fischer had no way of telling, so he guessed! He as- 
signed one arrangement, quite arbitrarily, to the d-glyceraldehyde 
and the other to the Z-glyceraldehyde, establishing this standard in 
a paper he published in 1891. 

(It wasn't till exactly sixty years later, in 1951, that it became 
possible to investigate molecules with sufficient subtlety to tell 
what the arrangement really was. This was accomplished by a team 
of Dutch investigators under J. M. Bijvoet, and they discovered 
that Fischer's fifty-fifty guess, unlike Franklin's was correct.) 

Fischer didn't stop there, of course. He began to build up, veiy 
carefully, more complicated sugar molecules, noting in every case 
what the arrangement must be. In every case, he could conclu- 
sively demonstrate that the structural arrangement of a compli- 
cated sugar with more than one asymmetric carbon atom was 
related to either the d-glyceraldehyde or the Z-glyceraldehyde 
standard. Provided the atomic arrangements in the standard com- 
pounds were as he guessed they might be, he could work out the 
arrangements of all the others. (If he guessed wrong, then he 
would have to switch the arrangement in every sugar - molecule to 
its mir ror image — but, as eventually turned out, he hadn't guessed 

He found that although d-glyceraldehyde was dextrorotatory, 
some of the compounds related to it, st A Hcturally, were levorota- 
tory. One could not predict from the structure alone the direction 
of optical rotation. Since lower-case letters had been used for di- 
rection of optical rotation, capital letters were used to indi- 
cate relationship. When a capital letter was used, the direction of 
rotation was indicated by (+) or ( — ), the former for dextro-, the 
latter for levo-. 

Thus, since the glucose found in living tissue is related to D- 
glyceraldehyde and is dextrorotatory, it is called D-(+)-glucose. 



The fructose found in living tissue is also related to D- 
glyceraldehyde and is levorotatory, so it is D-(-)-fructose. 

Here is something interesting. All the sugars found in 
living tissue, whether they turn the plane of polarized light in one 
direction or the other, are related to D-glyceraldehyde. They are 
all members of the "D-series." To put it more dramatically, the 
sugars of life are all right-handed.* 

But why? 

If we seek the reason for any regularity in the structure of com- 
pounds in living tissue, we are bound to look at enzymes. AH the 
compounds synthesized in living tissue are synthesized through 
the mediation of enzyme molecules, and all enzyme molecules are 

We must ask, then, as to the nature of the asymmetry of en- 

All enzyme molecules are proteins. Protein molecules are made 
up of chains of amino acids which come in some twenty varieties. 
All twenty varieties are closely related in structure. In each case 
there is a central carbon atom to which are attached: 1) a hy- 
drogen atom, 2) an amino group, 3) a carboxyl group, 4) any one 
of twenty different groups which may be lumped together as "side 

In the case of the simplest of the amino acids, "glycine," the 
side chain is another hydrogen atom, so that the central carbon 
atom is attached to only three different groups. For that reason, 
glycine is not asymmetric and is not optically active. 

In the case of all the other amino acids, the side chain repre- 
sents a fourth different group attached to the central carbon atom, 
which means that the central carbon is asymmetric and that each 
amino acid, except glycine, can exist in two forms, one the mirror 
image of the other. And, in fact, each amino acid exists in living 
tissue in only one of the two forms; and the same form is found, 
in each case, in all living tissue of any kind. 

* Minor exceptions? A substance related to L-(-)-glucose is found in 


But which form? Some amino acids in the naturally occurring 
form are dextrorotatory and some are levorotatory, but you can't 
go by that. Instead, you must work out their structural nature 
with reference to the glyceraldehyde standard. 

When this is done, it turns out that, without exception, all nat- 
urally occurring amino acids in all living tissue of whatever kind 
are of the L-series.* 

We can therefore eliminate all questions as to why this form of 
some sugar (or other compound) exists in tissue and not its mir- 
ror image, and zero in on the amino acids. From them, every- 
thing else follows, so we can ask: Why are all the amino acids of 
the L-series? 

It isn't hard to answer why all the amino acids belong to the 
same series. When amino acids hook together to form a protein 
molecule, the side chains stick out on this side or that and some 
of them are very bulky. The protein molecules do not have room 
to spare for them. 

If the amino acid chain were to consist of both L-amino acids 
and D-amino acids, there would be frequent occasions when an L- 
amino acid would be immediately followed by a D-amino acid. In 
that case, the side chains would stick out on the same side and 
would, in many cases, seriously interfere with each other. If, on 
the other hand, the chain consisted of L-amino acids only, the 
side chains would stick out first to one side, then to the other, al- 
ternately. There would then be more room available and a protein 
molecule could more easily form. 

But the same thing would be true if the chain consisted of D- 
amino acids only. In fact, there is no reason to think that proteins 
consisting of D-amino acids only would be in any way different in 
form or function from those that now exist, that organisms made 
up of such D-proteins would be in any way inferior to those that 
now exist, that a whole ecology based on D-organisms would be 
in any way less viable than the system which does exist on Earth. 

* Well, almost. There are some D-series amino acids found in very spe- 
cialized locations, in the cell walls of certain bacteria, for instance. 



The question, therefore, arises: Why one rather than the other? 
Why has Earth developed an L-ecology, rather than a D-ecology? 

The simplest possible explanation (and therefore the one which 
is perhaps most likely to be true) is through the working of sheer 

In the lifeless primordial ocean, individual more complex 
molecules were steadily being built up out of less complex precur- 
sors thanks to energy sources such as the ultraviolet radiation of 
the Sun. Among these molecules being built up were L-amino 
acids and D-amino acids.* These come together to form chains, 
such chains being built up most easily out of all one form or all 
the other, so that both D-chains and L-chains would exist. 

Eventually, some chains would be complex enough to have en- 
zymatic properties and could co-operate, perhaps, with nucleic 
acids that would also be forming. (Nucleic acids contain five- 
carbon sugars in their molecules, which are always of the D- 
series.) It may be that, through sheer circumstance, an L-amino 
acid chain was first to reach the necessary complexity and, in com- 
bination with nucleic acid, began multiplying. (It is characteristic 
of life that it is based on molecules capable of forming replicas 
of themselves.) 

In that way, the proto-life molecule, using itself as a model, 
could form many times more L-amino acid chains than could be 
formed by chance alone. The L-ecology would have got the first 
foothold and, being self-perpetuating, would never let go. The de- 
cision between L and D would thus be made at the very beginning 
of the history of life. 

It might just as well have gone the other way, too, so that if 
we were to study many Earthlike life -bearing planets, we might 
find that about half of them bore a D-ecology and half an L- 

Since food from D-organisms could be digested and assimilated 

* Since 1951. chemists have been trying to duplicate primordial condi- 
tions and have formed amino acids in this fashion — but always the D- and 
L-forms in equal quantities. 



only with difficulty, if at all, by I A organisms such as ourselves, 
and since it might set up serious, or even fatal, allergic manifesta- 
tions, human exploration of the Galaxy might then face a particu- 
lar danger. A planet might be a very paradise but if its life forms 
tested out D it would be unsuitable for colonization. 

But need we rely on pure randomness? There are some non- 
life sources of asymmetry. There is a kind of polarized light, called 
"circularly polarized light," which can be viewed as either a left- 
handed screw or a right-handed screw. 

A particular variety of such light, being asymmetric, would af- 
fect one minor-image compound more than its twin. A chemist 
beginning with an equal mixture of the two mirror images would 
end with one slightly in excess. He would go from symmetry to 
asymmetry without the intervention of life. Usually, though, he 
ends with only some 0.5 per cent of the amount of asymmetry he 
would get if he had one of the images only. 

Still, one can imagine a source of circularly polarized light on 
the primordial Earth, say through the reflection of sunlight from 
the ocean surface. The light might be harder on the D-amino acids 
than on the L-amino acids. The D-amino acids would be harder 
to form and easier to break down once formed. In that case there 
would be a kind of built-in bias in favor of the L-ecology. 

The catch is, though, that there seems no reason why the cir- 
cularly polarized light should be formed left-handed rather than 
right-handed. If it is formed in both ways equally, as is to be ex- 
pected, there will be no bias. 

But something new has turned up. 

A Hungarian botanist named Garay (I don't have his first 
name) reported in 1968 that an amino acid solution bombarded 
with energetic electrons from strontium-90, did not decompose 
equally. The D-forrn decomposed perceptibly more quickly than 
the L-forrn. 


One possibility is this. When the beta particles are slowed down 
by passage through the solution, they emit circularly polarized 


gamma rays. If the gamma rays were produced in equal amounts 
of left-handed and right-handed forms this wouldn't matter, but 
are they? 

As I explained in Chapter 2, the law of parity breaks down in 
weak interactions, and it is these which involve the electron. The 
breakdown means that the electron is not symmetric with respect 
to right and left. It is left-handed, so to speak. Consequently, the 
gamma rays it produces are left-circularly polarized and that means 
D-amino acids are less easily formed and more easily destroyed 
once formed. 

It would follow, then, that because of the non-conservation of 
parity there is an ingrained bias as far as optical isomers are con- 
cerned. In any Galaxy (or universe) made up of matter, in which 
electrons and protons dominate, we may expect a certain prepon- 
derance of L-ecologies among the life-containing planets. 

On the other hand, in any Galaxy (or universe) made up of 
antimatter, in which positrons and antiprotons dominate, we may 
expect preponderance of D-ecologies among the life-containing 

Of course, this postulated connection between non-conservation 
of parity and the asymmetry of life is, as yet, highly tentative, but 
I am emotionally drawn to it. I firmly believe that everything in 
the universe is interconnected, that knowledge is one; and it 
seems dramatically right to me to have a discovery concerning 
the non-conservation of the law of parity, which seems so ivory- 
towerish and far-removed, serve to explain something so funda- 
mental about life, about man, about you and me. 

B — The Problem of Oceans 


Cocktail parties bring out the worst in me in the way of self- 
righteousness, for I don't drink. 

This isn't a question of morality, you understand. It's just that 
I don't particularly like the taste of liquor and that even small 
quantities induce blotches and shortness of breath. Anyway, with- 
out ever touching a drop, I can be as hilariously drunk as anyone 
in the room — and no hangovers afterward. 

The only trouble is that people won't let it go at that. They 
stand around and hound me. "Are you sure you won't have some- 
thing?" they ask for the fifteenth time. 

What's more, when I do get thirsty, I have to go over to the bar- 
tender, make sure no one is listening, and then ask in a stage 
whisper if I can have some water. 

First, I have to convince him that I really want water. Then I 
have to persuade him that I want a large glass without ice. I gen- 
erally fail. Not listening, he picks out a cocktail glass and hands 
me water-on-the-rocks which means I have about five cubic centi- 
meters of fluid and must then stand there, moodily, swirling ice 
cubes and wishing they would melt. 

It's no wonder I get nasty. The other evening at a cocktail party 
one of those present was inveighing against marijuana. "Ninety- 
two per cent of heroin users," he said, "began with pot." 

I was on his side, actually, for I am against the use of drugs, but 
I eyed the glass of liquor he was holding and said, "Are you a so- 
cial drinker?" 

"Of course," he said. 

"Well," said I, "every single alcoholic who ever existed began 
as a social drinker." 


Anyway, there's nothing wrong with water. It's a great bever- 
age and a very unusual substance in addition. 

For instance, the six most common elements in the universe as 
a whole are thought to be hydrogen, helium, oxygen, neon, nitro- 
gen, and carbon, in that order. Out of every ten thousand atoms in 
the universe about 9,200 are hydrogen, 790 are helium, 5 are oxy- 
gen, 2 are neon, 2 are nitrogen, and 1 is carbon. All the rest make 
up an insignificant scattering and for many purposes can be simply 

With this information on hand, we can ask ourselves what the 
most common compound (i.e., a substance with a molecule made 
up of two or more different kinds of atoms) in the universe is. It 
stands to reason that the most common compound would be one 
with a small, very stable molecule made up of atoms of the two 
most common elements. 

Since helium atoms don't form parts of any molecules at all, 
that leaves hydrogen and oxygen as the most common compound- 
forming elements in the universe. One atom of each can combine 
to form "hydroxyl" (OH), which has been detected in the inter- 
stellar spaces of our galaxy and of at least one other. It can only 
exist in rarefied media such as that in space. Two hydrogen atoms 
and one oxygen form water (H 2 O), and that can exist at 
planetary densities — and is undoubtedly the most common such 
compound in the universe. 

Naturally, water wouldn't be common everywhere. It wouldn't 
exist at all in any normal star, of course. The molecule breaks up 
at stellar temperatures. On too-small planetary bodies, water mole- 
cules would be too light and flitting to be held by the feeble gravi- 
tational force. Some might be held by chemical forces to the rocky 
crust, but this would represent a very small percentage of the total 
potential. It is not surprising then that the Moon, Mars, and, un- 
doubtedly, Mercury are relatively dry. 

On giant planets such as Jupiter and Saturn, where the gravita- 
tional field is intense and the temperature is low, there is a much 
more representative sampling of the material of the universe and 



surely water is by far the most common compound on such worlds. 

Earth stands in an intermediate position. It is small enough and 
warm enough to have lost most of the water it might have pos- 
sessed at the start. More likely, it failed to gather most of it in the 
first place out of the swirling cloud of dust and gas from which the 
planet formed. Even so, water on Earth is extremely plentiful. 

In fact, in two respects. Earth's water is absolutely unique. In 
the first place, water is by far the most common liquid on Earth. 
Indeed, it is the only liquid on Earth present in quantity. (What 
is in second place? Petroleum, perhaps.) 

Secondly, water is the only substance on Earth present, in quan- 
tity, in all three phases, solid, liquid, and gas. Not only is there an 
ocean full of water, but there are polar caps of miles-deep ice, and 
there is water vapor making up a major (if variable) part of the 

The question. Gentle Readers, is this, then: Can any substance 
other than water serve? Can a planet exist with a large ocean of 
any substance other than water? 

To answer that question, let's consider the requirements: 

1) The ocean substance must be a plentiful component of the 
universe mixture. We can imagine oceans of liquid mercury, or 
liquid fluorine or liquid carbon tetrachloride, but we can't realis- 
tically imagine any planet with these particular substances present 
in such quantities as to spread out into oceans. 

2) The ocean substance must have a prominent liquid phase. 
For instance, the Martian polar caps may well be frozen carbon 
dioxide, but there is no liquid carbon dioxide phase at Martian at- 
mospheric pressure. The solid carbon dioxide vaporizes directly to 
gas, so there would be no carbon dioxide ocean even if there were 
enough carbon dioxide to form one. 

3) Ideally, we would want a substance whose liquid phase could 
be transformed with reasonable ease to either solid or gas, if we 
are to make possible those properties of Earth's ocean which lead 
to ice caps, clouds, rain and snow. Thus an ocean of liquid gallium 
at the temperatures of water's boiling point, for instance, might 
produce gallium "ice caps" with ease, but at that temperature, gal- 



Hum's vapor pressure would be so low that there would be no gal- 
lium vapor in the air to speak of, no gallium clouds, no gallium 
rain. On the other hand, if we had an ocean of liquid helium at a 
temperature of 2° above absolute zero (i.e., 2° K.) there would 
be plenty of helium vapor in the atmosphere (indeed, that would 
make up almost all the atmosphere) and helium rain would be 
common, but there is likely to be no helium ice or snow because 
solid helium doesn't form, even at absolute zero, except under con- 
siderable pressure, and we would be hard put to design a planet 
with sufficient atmospheric pressure at 2° K. to do the job. 

In considering the requirements, let's begin with the first- 
presence in oceanic quantities. For that, we had better work with 
the top six elements only: hydrogen, helium, oxygen, neon, nitro- 
gen, and carbon. Any substance made up of anything but these 
six elements (singly or in combination) might have many virtues 
but would simply not be present in sufficiently overwhelming a 
quantity to make up an ocean composed entirely or nearly en- 
tirely of itself* 

Of these six elements, two, helium and neon, can exist in ele- 
mental form only. A third, hydrogen, can form compounds, but 
exists in such overwhelming quantities that on any planet capable 
of collecting more than a trace of it (i.e., on Jupiter, as opposed to 
Earth) it must exist mostly in elemental form for sheer lack of 
sufficient quantities of other elements with which to combine. 

As for oxygen, nitrogen, and carbon, these, in the presence of 
a vast preponderance of hydrogen will exist only in combination 
with as much hydrogen as possible. Oxygen will exist as water 
(HtO); nitrogen, as ammonia (H 3 N); and carbon, as methane 

This gives us our list of the six possible thalassogens:** hydro- 

* There is one conceivable exception on an Earthlike planet. Silicon di- 
oxide is present in oceanic quantities but it is a solid and wouldn't be a 
liquid under anything but white heat. Scratch silicon dioxide. 

**This is a word I have just made up. It is from Greek words ("sea- 
producers") and I define it as "a substance capable of forming a planetary 



gen, helium, water, neon, ammonia, and methane, in order of de- 
creasing quantity. 

The next step is to consider each in connection with its liquid 
phase. At ordinary pressures, equivalent to that produced by 
Earth's atmosphere, each has a clear-cut boiling-point temperature, 
above which it exists only as a gas. This boiling point can be in- 
creased when pressure is increased, but let's ignore that compli- 
cation, and consider the boiling point, in degrees above absolute 
zero, at ordinary pressure. 

It turns out that the boiling points of helium, hydrogen, and 
neon are, respectively, 4.2° K., 20.3° K, and 27.3° K. 

But keep in mind that even distant Pluto has a surface tempera- 
ture estimated to be roughly 6o° K. In fact, I wonder if any siz- 
able planet, such as the outer members of our solar' system, can 
ever have extremely low temperatures. Internal heat arising from 
radioactivity must be sufficient to keep the surface temperature 
at Plutonian levels, at least, even in the complete absence of any 
sun. (Jupiter, for instance, according to a recent report I've 
seen, radiates three to four - times as much heat as it receives from 
the Sun.) 

In short, then, for any reasonable planet we can design, the 
temperature is going to be too high for the presence, in quantity, 
of helium, hydrogen, or neon in the liquid phase. Scratch them 
from the list and we have only three thalassogens left: methane, 
ammonia, and water. 

And what are their boiling points? Why, respectively, 111.7° K., 
239.8, and 373.2. 

If we consider these three, we come to these conclusions: 

1) Water is the most common and is therefore the most likely 
to form an ocean. 

2) Since methane is liquid across a range of 23 degrees, am- 
monia across 44 degrees, and water across 100 degrees, water, of 
the three, has by far the broadest temperature range for the liquid 
phase and, in its ocean-forming propensities, is least sensitive to 
temperature deviation. 

3) Most important of all, water forms its oceans at a higher 


temperature than the other two. You might expect methane 
oceans on a planet like Neptune or ammonia oceans on a planet 
lik e Jupiter. Only water, however, only water, could possibly form 
an ocean on an inner planet like Earth. 

Well then, we depend for the existence of our ocean, and there- 
fore for the existence of life, on the fact that water happens to 
have its liquid range at a far higher temperature than that of any 
other possible thalassogen. Is that just the way the ball bounces or 
is there something interesting to be wrung out of the water 

Let's see- 

When atoms combine to form molecules, the bond between 
them is formed through a kind of tug-of-war over the outermost 
electrons in those atoms. In many cases, one type of atom has the 
capacity to hold on to one or two electrons over and above those 
it normally possesses. Given half a chance it will grab on to such 
electrons. Since the atom itself is electrically neutral (positive 
charges in the interior, balancing negative charges on the out- 
skirts) and since eveiy electron has a negative charge, an atom 
which is capable of taking on one or more additional electrons 
then canies a net negative charge. Elements made up of atoms 
capable of doing this are therefore characterized as "electro- 

The most electronegative of the elements, by far, is fluorine. 
Following it, in order, are oxygen, nitrogen, chlorine, and bro- 
mine. These are the only strongly electronegative elements. 

Some atoms on the other hand have no strong ability to latch 
on to additional electrons. Indeed, they find it d if ficult to hold 
on to the electrons they normally possess and have a considerable 
tendency to give up one or two. Given half a chance they will do 
so. Once they lose such negatively charged electrons, what remains 
of the atom has a net positive charge. Such atoms are therefore 

Most of the elements tend to be somewhat electropositive. The 
most electropositive elements are the alkali metals, of which 


sodium and potassium are the most common representatives. Cal- 
cium, magnesium, aluminum, and zinc are other examples of 
strongly electropositive elements. 

When an electropositive element, like sodium, meets an electro- 
negative one like chlorine, the sodium atom freely gives up an 
electron, which the chlorine atom as freely takes. What is left is a 
sodium atom with a positive charge (a sodium ion) and a chlorine 
atom with a negative charge (a chloride ion). The attraction be- 
tween the two ions is the strong pull of an electromagnetic force 
and this is called "electrovalence." A number of chloride ions clus- 
ter around each sodium ion and a number of sodium ions cluster 
around each chloride ion. The result is an intricate and very or- 
derly array of ions that hang on to each other tightly. 

The commonest way of pulling ions apart is to use heat. All 
ions, no matter how firmly held in place by some sort of attrac- 
tion, are vibrating about that place. This vibration is related to 
temperature. The higher the temperature, the more energetic the 
vibration. If the temperature is high enough, the vibration be- 
comes violent enough to pull the ions apart, however strong the 
electromagnetic force between them, and the substance then 
melts. (In the liquid phase, the ions are no longer held firmly in 
place and they move about freely.) 

Nevertheless the temperature, by ordinary standards, must be 
quite high before the strong attractions between the sodium ions 
and the chloride ions can be overcome. Sodium chloride (ordinary 
table salt) has a comparatively high melting point, therefore — 
1074° K. (For orientation, a pleasant spring day with the tem- 
perature at 70° F. is at 294° K.) 

Still higher temperatures are required to pull the ions apart al- 
together and send them in pairs (one sodium ion and one chloride 
ion) into the nearly total independence of the gas phase, so that 
the boiling point of sodium chloride is 1686° K. 

This is more or less true of all electrovalent compounds, which 
form by the transfer of one or more electrons from one atom to 
another. Molybdenum oxide has a melting point of 2893° K. and 
a boiling point of 5070° K. 


What happens, though, when one electropositive element 
meets another? Sodium atoms, for instance, can form bonds 
among themselves by allowing the outermost electron each pos- 
sesses (and which they hold on to only veiy loosely) to be shared 
among them all. This is a stabler situation than would exist if each 
were responsible for its own outermost electron only, as in sodium 
gas. Consequently sodium atoms cling together and sodium is a 
solid at ordinary temperatures. To be sure, it doesn't take much to 
pull the atoms apart and sodium melts at a temperature of 370° K., 
just under that of boiling water. It doesn't boil, though, and ob- 
tain complete atomic independence till 1153 ° K. 

(Those outermost electrons wander easily from atom to atom. 
Their existence accounts for the fact that sodium, and metals gen- 
erally, conduct heat and electricity so much better than non- 

Metals made up of less electropositive atoms get together more 
snugly and some of them end up by forming bonds as tight as 
those of any electrovalent compound. Tungsten metal has a melt- 
ing point of 3640° K. and a boiling point of 6150° K. 

Yet though metallic atoms fit together well, there is a greater 
tendency for them to transfer electrons to the electronegative 
atoms, particularly to oxygen, which is by far the most common of 
all the strongly electronegative elements. For this reason, there is 
virtually no free metal in the Earth's crust.* 

In general, then, we can say that metals and electrovalent com- 
pounds are so high-melting as to offer no chance of a liquid phase 
at any reasonable planetary temperature, up to and including that 
of Mercury. Those few which might (like sodium metal or tin 
tetrachloride) cannot possibly be present in large enough quan- 
tity to form an ocean. 

So we must look for something else. What happens if one elec- 
tronegative atom meets another? What happens if one fluorine 

* Earth has a metallic core because it contains so much iron that there 
just aren't enough electronegative atoms to take care of it all. The metallic 
excess, denser than the oxygen-containing electrovalent compounds, settled 
to the Earth's center in the soft, youthful days of the planet. 


atom meets another, for instance? Each of the fluorine atoms can 
handle one electron over and above its usual assignment, but 
neither is in a position to give up one of its own in order to satisfy 
the other. What does happen is that each atom allows the other a 
share in one of its own electrons. There is a two-electron pool to 
which each contributes and in which each shares. Both fluorine 
atoms are then satisfied. 

In order for this pool to exist, though, the two fluorine atoms 
must remain at close quarters! To pull them apart takes a lot of 
effort, for it means breaking up that two-electron pool. Conse- 
quently, under ordinary circumstances, fluorine in elementary 
form exists in molecules made up of atom pairs (F 2 ). The tem- 
perature must rise well over 1300° K. even to begin to break up 
the fluorine molecule and shake the individual atoms apart. The 
attraction between atoms represented by shared electrons is called 
a "covalent bond." 

Two fluorine atoms, once they have formed their two-electron 
pool, have no reason to share any electrons with any other atoms, 
much less transfer electrons to them or even receive electrons from 
them. The two-electron pool completely satisfies their electron 
needs. Consequently, when one fluorine molecule meets another 
fluorine molecule, they bounce off each other, with very little tend- 
ency to stick together. 

If there were no tendency to stick together at all, the fluorine 
molecule would remain independent of its neighbors however far 
down the temperature might drop. The molecules would move 
more and more sluggishly, bounce off one another more and more 
feebly, but they would never stick. 

However, there are what are called "Van der Waals forces," 
named for the Dutch chemist who first studied them. Without 
going into the matter in detail we can simply say that there are 
weak attractive forces between atoms or molecules even when 
there is no outright electron transfer or electron sharing. 

Thanks to Van der Waals forces, fluorine molecules are slightly 
sticky, and if the temperature drops low enough, the energy that 


keeps them moving will not be great enough to make them 

break away after colliding. Fluorine will condense to a liquid. 

The boiling point of liquid fluorine is 85° K. If the temperature 
drops further still, the fluorine molecules lock firmly into an or- 
derly array and fluorine becomes a solid. The melting point of 
solid fluorine is 50° K. 

The same thing happens with the other electronegative ele- 
ments. Chlorine, oxygen, and nitrogen also form electron pools 
between two atoms. We therefore have chlorine molecules, oxy- 
gen molecules, and nitrogen molecules, each made up of atom 
pahs (Cl 2 , 0 2 , and N 2 ). Even hydrogen atoms, which are not par- 
ticularly electronegative, form molecules by pahs (H2). 

In eveiy case the melting and boiling points are low, with the 
exact value depending on the strength of the Van der Waals forces. 
Hydrogen, with its veiy small atoms, possesses a liquid range at a 
considerably lower temperature than that of fluorine. The boiling 
point of liquid hydrogen is 2 1 n K. and the freezing point of solid 
hydrogen is 14° K. 

A few varieties of atom happen to possess a satisfactory number 
of electrons to begin with. They have little tendency to give up 
any electrons they have and still less to accept additional electrons 
from outside. They do not therefore tend to form compounds. 
These are the so-called "noble gases." 

There are six of these altogether and, of them, the three with 
the largest atoms can form compounds (not veiy stable ones) 
with the most electronegative elements, such as fluorine and oxy- 
gen. The three with the smallest atoms — argon, neon, and helium 
(in order of decreasing size) — won't do even that much under 
any conditions yet discovered. Nor will they form electron pools 
among themselves. They remain in sullen isolation as individual 

Yet they, too, experience the mutual attraction of Van der 
Waals forces and, if cooled sufficiently, become liquids. The smaller 
the atom, the smaller the forces and the more strongly cooled 
they must be to liquefy. 

Helium, with the smallest atoms of the noble gases, experiences 


such small attractions that of all known substances it is the most 
difficult to liquefy. The boiling point of liquid helium is phenome- 
nally low, only 4.2° K. Solid helium doesn't exist at all, even at o° 
K. (absolute zero), except under considerable pressure. 

So far, though, these gaseous substances I have discussed, that 
are covalent in nature, and that have liquid ranges far down the 
temperature scale, are all elements — elements that either exist in 
the form of isolated atoms, as in the case of helium, or as isolated 
two-atom molecules, as in the case of hydrogen. 

Is it possible for molecules of two different atoms to be covalent 
in nature and to be low-melting and low-boiling. — Yes, it is! 

Consider carbon. The carbon atom is neither strongly electro- 
positive nor strongly electronegative. It has a tendency to form 
two-electron pools with each of four other atoms. It could form 
those pools with four other carbon atoms, each of which can 
form pools with three others, each of which with still three others, 
and so on indefinitely. In the end, uncounted trillions of carbon 
atoms may be sticking firmly together by way of strong covalent 
bonds. The result is that carbon has a higher melting point than 
that of any other known substance — nearly 4000° K. 

But the carbon atom may form a two-electron pool with each 
of four different hydrogen atoms. The hydrogen atoms can only 
form one two-electron pool apiece and so that ends it. The entire 
molecule consists of a carbon atom surrounded by four hydrogen 
atoms (H4C), and this is methane. 

Methane molecules have little attraction for each other except 
by way of weak Van der Waals forces. The boiling point of liquid 
methane is 112° K., and the melting point of solid methane is 
89° K. 

Similarly, a carbon atom can form a molecule with one oxygen 
atom. This would be carbon monoxide (CO). Its boiling point 
and melting point are, respectively, 83 0 K. and 67° K. 

Now we can come to a general conclusion. Unlike metallic sub- 
stances and electrovalent compounds, covalent compounds have 



low melting points and boiling points and only they can conceiv- 
ably be thalassogens at reasonable planetary temperatures. 

This gives us our first answer as to why water is a thalassogen 
at all: it is a covalent compound essentially. All right, that's some- 
thing to begin with. Yet so many covalent compounds are, if any- 
thing, liquid at too low a range for planetary purposes and 
certainly for earthly purposes specifically. Why is liquid water so 
warm then? 

One possibility rests in the fact that, in general, the larger the 
covalent atom or molecule, the stronger the Van der Waals forces 
and the higher the boiling point. Consider the following table, in 
which the size of the molecule is measured by its molecular weight 
(or, in the case of helium and neon, atomic weight). 


Atomic or Molecular 

Boiling Point 


Hydrogen (H 2 ) 



Helium (He) 



Neon (Ne) 



Nitrogen (N 2 ) 



Carbon monoxide (CO) 



Oxygen (O a ) 



Fluorine (F 2 ) 



Oxygen fluoride (OF 2 ) 



Nitrogen fluoride (NF 3 ) 



Chlorine (Cl 2 ) 



Pentane (CsHi 2 ) 



Chlorine heptoxide (C 1 2 0 7 ) 



The table isn't perfect, for helium, which has a larger atomic 
weight than hydrogen's molecular weight, nevertheless has a lower 
boiling point than hydrogen. Then, too, fluorine, which has a 
larger molecule than oxygen has, is nevertheless lower-boiling. 
Still, the table seems to show that there is a kind of rough and 
ready relationship between molecular weight and boiling point in 
the case of covalent compounds. 



We might conclude therefore that water, which has a boiling 
point at 373° K., ought to have a molecular weight somewhat 
higher, or at least not particularly lower, than chlorine heptoxide. 
Its molecular weight ought to be, say, 180, as a minimum. 

Except that it isn't. The molecular weight of water is 18, just 
one tenth what it "ought" to be. 

Something, obviously, is terribly wrong — or right, perhaps, for 
it is to whatever causes this anomaly that we owe our life-giving 
ocean. What that wrongness/rightness might be we'll discuss in 
the next chapter. 


One of the occupational hazards of popularizing the scientific view 
of the universe for the general public is the occasional collision 
with readers who prefer some variety of religious view of the uni- 
verse instead. To reduce some wonderful phenomenon from the 
provenance of God to the blind consequence of some physical or 
chemical "law" offends them, and their response, very often, is to 
accuse the science writer of atheism. 

Thus, only yesterday, I received a letter from a lady which began 
by addressing me austerely as "Dear Sir," and then continued, 
somewhat less austerely, "According to the Scriptures, and using 
the language of the Scriptures, you are a 'fool.'" 

That aggrieved me, naturally, since while I am every bit as fool- 
ish (on occasion) as the next fellow, I hate to be told so. Besides, 
the accusation went beyond that of mere folly. It was obvious that 
the lady was referring to a certain well-known biblical quotation. 

Among the hundred and fifty poems in the Book of Psalms, 
there are two, the fourteenth and the fifty-third, that are virtually 
identical, and in each case the first verse begins, "The fool hath 
said in his heart. There is no God." 

What could I do? I decided that a scriptural reference deserves 
a scriptural reference, so I sent the nice lady a short note which 

. . whosoever shall say. Thou fool, shall be in danger of hell 
fire.' Matthew 5:22."* 

But, alas, having taken care of one correspondent, I must now 
run the risk of offending others of those whom Robert Burns 
would refer to as the "unco guid." For, you see, water has amazing 

* This is from the Semion on the Mount, in case you don't recognize it. 


properties that seem to be perfectly designed for life. It would be 
so pious to look upon it as the workings of a benevolent and in- 
genious Maker, creating a Universe for the good of undeserving 
Man, and so prosaic to bring it down to the uncaring properties 
of atoms. 

— Yet I will have to do the latter, since I am committed to the 
scientific view of the universe (pointing out to the reverent that 
they can easily suppose those uncaring properties to have been 
created by God). 

In the previous chapter, I pointed out that water was the only 
possible thalassogen for a planet at the temperature of the Earth, 
the only compound that could possibly exist in sufficient quantity 
in the liquid phase to form an ocean. 

To be liquid at the relatively low temperatures of Earth (as I 
explained), a substance would have to consist of covalent mole- 
cules, that is, molecules in which pairs of neighboring atoms more 
or less share electrons in neighborly fashion, rather than carrying 
through a transfer of one or more electrons from one atom to 
another, bodily. 

In general, the larger the molecular weight of a covalent com- 
pound the higher the temperature range of its liquid phase. From 
that standpoint one might expect a substance which is liquid at 
water temperatures to have a molecular weight of perhaps 180. 
The molecular weight of water, however, is 18, just one tenth of 
what it "ought" to be. On the basis of molecular weight, liquid 
water is surprisingly warm; it is "hot water" indeed. 

But why is that? Are we perhaps oversimplifying if we relate 
liquid-phase temperatures to molecular weight alone? 

Well, at the end of the previous chapter I listed molecular 
weights and boiling points without any attempt to pick and choose 
among them. That is probably unfair, for substances are made up 
of different elements, and these differ greatly among themselves in 
chemical and physical properties. Yet elements exist in families, 
and within these, the members are quite similar. It might be best 



to stick to members of a particular family and see what regularities 
we can find there. 

For instance, consider the six elements of the noble gas family, 
their atomic weights, and their boiling points. 

Table 1 


Atomic Weight 

Boiling Point 

Helium (He) 



Neon (Ne) 



Argon (Ar) 



Krypton (Kr) 



Xenon (Xe) 



Radon (Rn) 



Here we have a smooth rise in boiling point with the atomic 
weight, which is what we would expect, looking at the matter in 
an unsophisticated way. After all, if the atoms grow heavier, it 
takes more energy in the form of heat to lift them away from each 
other and send them off separately in vapor form. 

What if we shift to the four elements of another family, the 
halogens, a family as well defined as that of the noble gases (see 
Table 2). Here, too, the boiling point rises smoothly with the 

Table 2 


Atomic Weight 

Boiling Point 

Fluorine (F) 



Chlorine (Cl) 



Bromine (Br) 


33 1( ? 

Iodine (I) 



atomic weight. There is a fifth halogen, the last in the series, which 
is named astatine. It is a radioactive element and even its most 
long-lived nuclear variety (with an atomic weight of 210) has a 

*"° K." represents the "absolute scale of temperature with the zero 
point ("absolute zero") at — 273.16° C. 



half-life of but 8.3 hours. It has not yet been obtained in quantities 
large enough to allow a clear boiling point determination, but 1 
am willing to bet, sight unseen, and any reasonable sum, that its 
boiling point is somewhere in the neighborhood of 570 n K. 

While the progression is smooth within an element family, ob- 
serve what happens when we cross the line. Compare Tables 1 and 
2. Neon and fluorine are not very different in atomic weight but 
fluorine's boiling point is three times as high as neon’s. This goes 
all the way down, each halogen having a boiling point approach- 
ing three times that of the noble gas with similar atomic weight. 

Is it that atomic weight alone isn't the sole deciding factor? Of 
course not. There are other properties that play a role. The noble 
gas atoms are chemically inert and do not combine among them- 
selves. They remain as separate atoms. The halogen atoms, on the 
other hand, because of their characteristic electron arrangements 
(different horn those of the noble gas atoms) do combine in pairs. 
Fluorine is not composed of individual atoms, as neon is, but of 
molecules made up of two atoms apiece. Fluorine is F2 and its 
molecular' weight is 38.0. To consider the energy required to sepa- 
rate the fundamental particles of fluorine liquid into vapor, one 
ought to consider the weight of the molecule, not that of the atom. 
The molecular - weight of fluorine is about that of the atomic weight 
of argon and, sure enough, the boiling point of fluorine is about 
that of argon. 

If we could stop there, we'd be able to work up a hard-and-fast 
relationship between particle size (whether atomic or molec- 
ular) and boiling point. In science, though, it isn't fair' to stop 
at any point where you find yourself with the answer you want. 
You have to be sporting enough to look further and try to spoil 
your - own hypothesis. 

That's not har'd to do. Chlorine atoms combine by twos also, 
and chlorine is CF, with a molecular' weight of 71. That is dis- 
tinctly less than the atomic weight of krypton, and yet the boiling 
point of chlorine is just twice that of krypton. 

So we had better not try to cross the family lines in working 
up our - theories. For the rest of the article I will stick to families 



and it will only be anomalies within the families that will receive 
our attention. 

But let's see, is it only the boiling points that vary smoothly 
with atomic (or molecular) weight? Is the variation always direct, 
so that the measure grows larger as the weight goes up? Let's con- 
sider a third well-marked element family, that of the "alkali met- 
als" and take their melting points this time. 

Table 3 


Atomic Weight 

Melting Point 

Lithium (Li) 



Sodium (Na) 



Potassium (K) 



Rubidium (Rb) 



Cesium (Cs) 

i3 2 -9 

3 01 

Cesium's melting point is down to 301 ° K., or 28.5° C, which 
means that it will melt on a hot summer day. There is also a sixth 
alkali metal, francium, which is radioactive, with its most long- 
lived nuclear variety (atomic weight, 223) having a half-life of only 
21 minutes. Its melting point has not been determined but you 
can bet, though, it is very likely about 290° K. and that it will 
melt on a balmy spring day. 

Other properties of other kinds vary in this regular fashion with 
atomic weights within element families, with values sometimes 
moving steadily upward and sometimes steadily downward.* The 
next question is, though, will this same sort of happy effect work 
within families of compounds — that is, substances with molecules 
made up of more than one kind of atom? 

Consider molecules made up of carbon and hydrogen. These 
come in many varieties, because carbon atoms can link together 
in chains and rings. Suppose, then, we consider a single carbon 
atom combined with hydrogen, a chain of two carbon atoms com- 

* It is only fair to say that a rigidly steady variation is not always found. 
There are exceptions. However, modern chemists can usually account for 
them, and we will have an example later in this article. 



bined with hydrogen, a chain of three carbon atoms, four, and 
so on. The longer the chain the larger the molecular weight, and 
we can consider such a series of progressively larger molecules of 
very much the same kind to make up a family. What happens to 

the boiling point, 

in that case? 

Table 4 


Molecular Weight 

Boiling Point (° K.) 

Methane (CH 4 ) 



Ethane (C 2 H 6 ) 



Propane (C 3 H 8 ) 



Butane (C4H10) 



Pentane (C5H12) 



Hexane (CeH A ) 



As you see, boiling point rises smoothly with molecular' weight 
in this case. 

To be sure, the family of "hydrocarbons" considered in 
Table 4 is one in which all the member's have molecules made up 
of the same elements. Would it be possible to set up families in 
which at least one of the elements changes from member to 

Thus, carrion is the first member of an element family of which 
the next three, in order of increasing atomic weight, are silicon 
(Si), germanium (Ge), and tin (Sn). An atom of each of these 
higher member's can combine with four' hydrogen atoms to form 
well-known compounds (silane, germane, and stannane, respec- 
tively) analogous to methane. Table 5 shows what happens to the 
boiling points there, and you see we get regularity in such a 
family, too. 

The problem, then, of finding out why water has the high 
liquid-range temperatures it has, may become easier to handle if 
we work within some family of compounds that includes it. 

Water molecules are made up of hydrogen and oxygen atoms 
(H 2 0). Of these two elements, hydrogen is a loner and is not 



Table 5 


Molecular Weight 

Boiling Point 

Methane (CH4) 



Silane (SiH 4 ) 



Germane (GeH 4 ) 



Stannane (S11H4) 



part of any clearly defined family (though it has certain relation- 
ship both to the halogens and to the alkali metals). Oxygen, on 
the other hand, is the first member of a family that includes sul- 
fur (S), selenium (Se), and tellurium (Te) as later members. 
An atom of each of these three can combine with two hydrogen 
atoms to form molecules (EES, H 2 Se, and H 2 Te, respectively) 
that are analogous in structure to water molecules. 

Table 6 




Boiling Point 
(° K.) 

Water (H 2 0) 



Hydrogen sulfide (H 2 S) 



Hydrogen selenide (H 2 Se) 



Hydrogen telluride (H 2 Te) 



If we look at the last three members alone, we see that the 
boiling point goes up with molecular weight. But water doesn't 
fit! Its boiling point should be, judging from the rest, something 
like 200° K. or — 73 ° C. Only the coldest polar days should suffice 
to liquefy its vapor and yet here it is, boiling something like 170 
degrees higher than it should. Hot water, indeed. 

There are two other compounds that, like water, don't fit their 
families in this respect. 

A hydrogen atom will combine with one atom of any of the 
halogens. We can get hydrogen fluoride (HF), hydrogen chloride 
(HC1), hydrogen bromide (HBr), and hydrogen iodide (HI). 



The boiling points of the last three on the absolute scale are 
188.2, 206.5, an d 237.8, respectively. We might expect HF to have 
a boiling point about 170, but it doesn't. Its boiling point is 
292.6 or about 120 degrees "too high." 

Then, too, three hydrogen atoms will combine with one atom 
of the member of a family of elements that includes nitrogen 
(N), phosphorus (P), arsenic (As), and antimony (Sb). The 
compounds phosphine (H3P), arsine (FFAs), and stibine 
(HiSb) have boiling points of 185.5, 218, and 256. On that basis, 
the first member of the series, ammonia (H3N), ought to have a 
boiling point of about 150, but it doesn't. Its boiling point is 
239.8, which is about ninety degrees "too high." 

What, then, do these three too-high-boiling compounds, water 
(H2O), ammonia (H3N), and hydrogen fluoride (HF), have in 

1) All three are made up of molecules consisting of hydrogen 
atoms and one other kind of atom. 

2) The other atoms involved — oxygen, nitrogen, and fluorine — 
just happen to be the three most electronegative atoms there are; 
that is, the atoms most capable of snatching electrons from other 

A fluorine atom, the most electronegative of all, can, for in- 
stance, take an electron away from a sodium atom altogether, 
assuming sole ownership and leaving the sodium atom utterly 
minus one electron. 

The hydrogen atom is not quite such an easy mark. It holds on 
to its single electron more tightly than the sodium atom does to 
its one outermost electron. The fluorine atom does not take 
hydrogen's electron away altogether, but it does take over the 
lion's share of it. The electron, so to speak, is closer to the center 
of the fluorine atom than to the center of the hydrogen atom. 

This means that if you imagine a line drawn down the center 
of the hydrogen fluoride molecule, with the hydrogen atom on 
one side and the fluorine atom on the other; the fluorine side, 
having more than its equal share of electrons, has what amounts 



to a small negative electric charge, while the hydrogen side has 
an equally small positive electric charge. 

Much the same can be said of the water molecule and the 
ammonia molecule. In each case, the side of the hydrogen atoms 
carries a small positive charge, while the side of the oxygen (or 
nitrogen) atom carries a small negative charge. 

All three molecules are "polar molecules." That is, they have 
poles at which electric charge is concentrated. 

This is not true of H 2 S, for instance, which is otherwise so 
similar to H20 in structure. Sulfur just isn't as electronegative as 
oxygen and it cannot hog more than its fair share of the electrons 
of the hydrogen atoms. Hydrogen sulfide is therefore not par- 
ticularly polar. Neither is hydrogen chloride or phosphine. 

If we now consider polar molecules, those with a positively 
charged end and a negatively charged end, we must inevitably 
stall thinking of the possibility of attraction between molecules. 
What if the positively charged end of one molecule should be 
near the negatively charged end of another molecule of the same 
kind? Would they not stick together a bit? 

Yes, they would, particularly since the positively charged end 
involves the hydrogen atom. Why? Because the hydrogen atom is 
the smallest of all the atoms and its center can therefore be most 
closely approached. The strength of attraction between two op- 
positely charged objects varies inversely as the distance between 
them. The closer they come together, the stronger the attraction. 

It follows, then, that the water molecule, the hydrogen fluoride 
molecule, and the ammonia molecule are "sticky molecules." They 
tend to line up positive end to negative end, and it takes signifi- 
cantly higher temperatures to piy them apart than if they were 
non-polar; that is, lacking the concentration of charge on two op- 
posite sides, and held together only by the Van der Waals forces 
mentioned in the previous chapter.* 

* Van der Waals forces are also the result of electrical asymmetry in atoms 
and molecules, with momentary concentrations of electric charge in one 
place or another. In non-polar molecules, however, the concentration-shifts 
from place to place lead to overall polarizations that are tiny indeed, much 



Usually, the water molecules are pictured with a hydrogen atom 
attached to the oxygen atom of its own molecule by a solid bond 
representing an ordinary chemical linkage, while it is attached to 
the oxygen atom of a neighboring molecule by a longer, dashed 
bond to indicate the electromagnetic attraction of opposite 

Because the hydrogen atom is thus between two oxygen atoms, 
one of its own and one of a neighboring molecule (or, in similar 
fashion, between two fluorine atoms, between two nitrogen 
atoms, between a nitrogen atom and an oxygen atom, and so on), 
the situation is commonly referred to as a "hydrogen bond." 

The hydrogen bond is only about one twentieth as strong as 
an ordinary chemical bond, but that is enough to add up to 170 
degrees to the temperature required to tear the molecules apart 
and set the liquid to boiling. Water molecules are sticky enough, 
thanks to the hydrogen bonds, to boil at 373 ° K. instead of 200° 
K., and that, combined with the fact that hydrogen and oxygen 
are the two most common compound-forming atoms in the uni- 
verse, makes it possible for oceans of liquid to exist on a planet the 
temperature of the Earth. 

What's more, it is because of the stickiness of the water mole- 
cules that it is possible for water to absorb so much heat for each 
degree rise in temperature or give off so much heat for each de- 
gree fall. We say, therefore, that water has an unusually high "heat 

There is, similarly, an unusually high heat absorption at the 
melting point or boiling point, due to the necessity for breaking 
all those hydrogen bonds. That is, it takes much more heat than 
one would expect to convert ice at 273 ° K. to water at the same 
temperature, or to convert water at 373 ° K. to steam at the same 
temperature. Working in reverse, an unusual amount of heat is 
given off when steam condenses to water or water freezes to ice. 
(Water has an unusually high "latent heat of fusion and vapori- 
zation," in other words.) 

less than those in polar molecules, where the concentration of charge is 
persistent and definitely localized. 


This is more than a mere matter of statistics. Water acts as a 
huge heat-sponge. It takes up and gives off more heat than any 
other common substance for a given change of temperature, so 
that the ocean rises in temperature much more slowly under the 
beating rays of the Sun than the land does, and drops in tempera- 
ture much more slowly in the absence of the Sun. 

With a vast ocean of water on its surface, the Earth therefore 
has a much more equable temperature than it would without it. 
In the summer, the sluggishly warming ocean acts as a cooling 
device; in the winter, the sluggishly cooling ocean is a warming 
device. And if you want to see what that means in a practical 
sense, consider the temperature ranges over the day-night interval 
and the summer-winter interval of a land area far from any 
ameliorating stretch of ocean (North Dakota) with those of one 
that is surrounded by ocean on all sides (Ireland). 

Since at any temperature, the evaporation of water absorbs 
more heat per gram of vapor formed than is true of any other 
common liquid, water is a particularly cheap and effective air- 
conditioning device. 

Perspiration is almost pure water and as it evaporates a great 
deal of heat must be absorbed from the object closest to that water 
— which happens to be the skin on which the perspiration rests. 
In this way, the body is cooled. 

Then, too, there is the matter of solvent properties. In a sub- 
stance like sodium chloride (common salt), the sodium atoms 
lose an electron each to the chlorine atoms, which therefore gain 
an electron each. The sodium atoms carry a unit positive charge 
and the chlorine atoms a unit negative charge, and are hence 
called ions. The two sets of ions cling together through the at- 
traction of opposite charges.* 

When particles of salt are dumped into water, the presence of 
positive and negative poles on the water molecules sets up an 
electromagnetic field which tends to neutralize that which is set 
up by the charged sodium and chloride ions. The ions cling to 

* The sodium chloride combination is much more polar than the water 
molecule is and this is reflected in its extremely high boiling point. 



each other with far less verve in the presence of water than in the 
open air and have a pronounced tendency to fall apart and go 
swimming in the water on their own. To put it briefly, sodium 
chloride dissolves in water. 

So do a surprising variety of other electrovalent compounds, 
that is, compounds made up of oppositely charged ions after the 
fashion of sodium chloride. 

Polar compounds, which are not built up of outright ions but 
have molecules with separated concentrations of charge (like 
water itself), also lose a considerable part of their tendency to 
cling together in the presence of water and therefore tend to 
dissolve. This includes many common substances of importance 
to life which have the oxygen-hydrogen or nitrogen-hydrogen 
linkage that makes polarization possible. 

This includes various alcohols, sugars, amines, and other or- 
ganic compounds. 

No other liquid is so versatile a solvent as water; no other liquid 
can dissolve appreciable quantities of so wide a variety of sub- 
stances. To be sure, though, water cannot dissolve appreciable 
amounts of all electrovalent compounds, since electrovalency is 
not the only property that is important. And, of course, it cannot 
dissolve non-polar compounds such as hydrocarbons, fats, sterols, 
and so on. 

The importance of water's versatile solvent action is this — 

The body's most important substances, the proteins and the 
nucleic acids, together with its most important fuels, the starches 
and sugars, are loaded with oxygen-hydrogen and nitrogen- 
hydrogen linkages and, if not polar altogether, have important 
polar regions within their molecules. Such compounds can there- 
fore dissolve in water or, at least, can attach water molecules 
intimately to various portions of their structure and undergo 
changes in connection with these attached water molecules. 

In short, the body's chemistry can go on against the intimacy 
of a water background. This background is so essential to life as 
we know it, that life could only have reasonably begun in the 


ocean, and now, even where it has adapted itself to dry land, the 
tissues remain approximately 70 per cent water. 

So consider water. Consider its high liquid-range temperatures, 
its capacity to act as a temperature-ameliorating heat-sponge and 
as an efficient air conditioner, its ability to dissolve a wide variety 
of substances and, therefore, to act as a medium within which 
the reactions necessary to life can proceed, and you may well say, 
"Surely, this is no accident. Surely water is a substance that has 
been carefully designed to meet the needs of life." 

But that is placing the cart before the horse. I'm afraid. Water 
existed, to begin with, as a substance of certain properties, and 
life evolved to fit those properties. Had water had other properties, 
life would have evolved to fit those other properties. If water had a 
lower liquid-range temperature, for instance, life might have 
evolved on Jupiter. And if water had not existed at all, life might 
have evolved to fit some other substance altogether. 

In every case, though, life would have evolved so neatly to fit 
whatever was at hand, that any form of that life high enough to 
consider the situation with sufficient subtlety would well feel 
justified in believing that intelligent and purposeful supernatural 
design was involved in something which, actually, the blind and 
random forces of evolution had produced. 

And I suppose my delightful lady correspondent, if she were so 
hardy as to read through this essay carefully, would but feel her- 
self further justified in her belief about the relationship of scrip- 
tural language to myself. 

But what can I do? I call the situation as I see it. 


About half a year ago (as I write this) I was hurrying through 
the wintriness of New York City. There was no snow on the 
ground but it was cold and I was hastening for haven. As I was 
crossing the street, my foot came down upon a manhole cover and 
a fraction of a second later I had made hard and full-length con- 
tact with the ground. 

It was the hardest fall I had ever taken and my first thought, 
as I lay there, was one of regret, for it felt as though I had broken 
my left tibia and in all my thirty-plus years I had never before 
broken a bone. I ought to have lain there and waited for help, but 
I had to struggle to my feet for two reasons: 

For one thing, I was hoping desperately the bone was not 
broken and if I could get to my feet it wouldn't be. Secondly, I 
wanted to find out why I had fallen, since I am usually reasonably 

I found I could stand. My left leg was banged up below the knee 
but the bone was intact, even though my suit (my best suit) 
was not. I further found (more in anger than in sorrow) that the 
manhole cover was frosted over with a thin layer of slippery ice. 
What had laid me low was the fact that the ice was quite trans- 
parent and that without close inspection, the manhole cover 
seemed bare and safe. 

I had to hobble onward, at that moment, toward my hotel 
room, which was four infinitely long blocks away, and there was 
no time to muse on what had happened and make an article out 
of it. By now, though, the bitterness of the time has been some- 
what assuaged, and I am ready. So here, O Gentle Reader, is the 
result — 



To the ancients, one of the remarkable things about ice, per- 
haps the most remarkable, was the very property that had caused 
my near disaster — its transparency. To the Greeks, ice was 
krystallos from kryos, meaning "frost," so the first strong impres- 
sion seems to have been left by its manner of formation. 

Once that was established, however, another property super- 
vened and the word came to be more significant for the connota- 
tion of transparency than of cold. After all, anything at all could 
be cold, but in ancient times few objects were known that were 
at the same time solid but not opaque. 

It followed, then, that when pieces of quartz were discovered, 
and found to be transparent, they were called krystallos, too, and 
were considered (at first) to be a form of ice that had been sub- 
jected to such intense cold as to have attained permanent solidity 
and an inability ever to melt again. 

Then the word achieved still another change of connotation. 
One interesting fact about transparent quartz was its surprising 
regularity in shape. It had plane faces that met to form clearly 
defined angles and edges. Consequently, krystallos came to 
mean any solid with such a regular geometry. From this came our 
modern word "crystal." 

Nevertheless, the older meaning of transparency persists in 
vestigial fashion. One still hears of the "crystalline spheres" which 
held the planets in the old Ptolemaic cosmology. This was not 
because they consisted of solid crystals; heavens, no. It was be- 
cause they were perfectly transparent so they could not be seen. 

And in modern times, the fortuneteller, gazing mystically into 
a glass sphere, is pretending to see something in her "crystal ball." 
This is not because the sphere is crystalline in the modern sense, 
for glass happens to be one of the very few common solids that is 
not crystalline (and, therefore, not truly solid), but because it is 

And yet, none of that really represents the true wonder of ice. 
It may seem to have wonders enough. Its mere existence as "hard 
water" may seem amazing and paradoxical enough to the life- 



long inhabitants of tropic climes, and its coldness and trans- 
parency may be of interest, but all that is really nothing. 

Consider instead something that is often remarked on, to the 
point, in fact, where it becomes something of a cliche. Have you 
never heard a statement such as this one: "Like an iceberg, nine 
tenths of the significance of the remark was hidden"? 

Like an iceberg! 

Being a non-traveler, I have never seen a real iceberg, but if I 
were on a ship and one hove into view (at, I hope, a safe distance) 
I am sure that the passengers, crowding against the rail to see it, 
would say to one another, "Just imagine, Mabel (or Harry), nine 
tenths of that iceberg is under water." 

Then I would say, "That's not surprising, ladies and gentlemen. 
The surprising thing is that one tenth of that iceberg is above 
water." Naturally, that would mean I would start getting those 
queer looks that would indicate once again (oh, how many 
times!) how much of a nut I appear to be to my beloved fellow- 

But it's true- 

In general the density of any substance increases as tempera- 
ture goes down. The lower the temperature, the more slowly the 
atoms or molecules of a gas move, the less forcefully they bounce 
off each other, and the closer they can crowd to each other. When 
the kinetic energy of the gas molecules is insufficient to overcome 
the attractive forces between the molecules (see the previous two 
chapters), the gas liquefies. 

In liquids, the molecules are in virtual contact, but they have 
enough energy to slip and slide past each other freely. They also 
vibrate and keep each other at greater distances than would be 
the case if all were absolutely motionless. As the temperature 
drops, the vibrations decrease in force and amplitude and the 
molecules settle a bit closer together. The density continues to 

Eventually, the energy of vibration isn't enough to keep the 
molecules slipping and sliding. They settle into a fixed position 
and the substance solidifies. The settling is more compact than is 


possible (usually) in the liquid form but there is still vibration 
about the fixed position. As the temperature continues to drop, 
the vibrations continue to die down until they are reduced to a 
minimum at the temperature of absolute zero ( — 273.1° C). It is 
then that density is at a maximum. 

To summ a rize — As a general rule, there is an increase in 
density with decrease in temperature. There is a sudden sharp 
increase in density when a gas becomes a liquid* and another, but 
lesser, sharp increase when a liquid becomes a solid. This means 
that the solid foim of a substance, being denser than the liquid 
foim of that same substance, will not float in the liquid form. 

As an example, liquid hydrogen has a density of about 0.071 
grams per cubic centimeter, but solid hydrogen has a density of 
about 0.086 grams per cubic centimeter. If a cubic centimeter of 
solid hydrogen were completely immersed in liquid hydrogen it 
would still weigh 0.015 grams and would be pulled downward 
by gravity. Sinking slowly (against the resistance of the liquid 
hydrogen) but definitely, it would eventually reach the bottom 
of the container, or the bottom of the ocean, if there were that 
much liquid hydrogen. 

(You might suspect that the solid hydrogen would melt on the 
way downward, but not if the ocean of liquid hydrogen were at 
its freezing point — and we'll suppose it is.) 

In the same way, solid iron would sink downward through an 
ocean of liquid iron, solid mercury through liquid mercury, solid 
sodium chloride through liquid sodium chloride, and so on. This 
is so general a situation that if you took a thousand solids at 
random, you would be veiy likely to find that in each case the 
solid foim would sink through the liquid foim and you would be 
tempted to make that a universal rule. 

— But you can't, for there are exceptions. 

And of these, by far the most important one is water. 

At ioo° C. (water's boiling point under ordinary conditions), 
water is as un-dense as it can be and still remain liquid. Its density 

* Except at the "critical temperature," something which need not concern 
us now. 



then is about 0.958 grams per cubic centimeter. As the tempera- 
ture drops the density rises: 0.965 at 90° C, 0.985 at still lower 
temperature, and so on until at 4° C, it is 1.000 grams per cubic 

To put it another way, a single gram of water has a volume of 
1.043 cubic centimeters at 100° C, but contracts to a volume of 
1.000 cubic centimeters at 4° C. 

Judging bom what is true of other substances, we would have 
every right to expect that this increase in density and decrease in 
volume would continue as the temperature dropped below 4° C. 
It does not! 

The temperature of 4° C* represents a point of maximum 
density for liquid water. As the temperature drops below that, the 
density stalls to decrease again (veiy slightly, to be sure) and by 
the time one reaches o° C, the density is 0.9999 grams per cubic 
centimeter, so that a gram of water takes up 1.0001 cubic centi- 
meters. The difference in density at o° C. as compared with that at 
4° C. is trifling, but it is in the "wrong" direction, and that makes 
it crucial. 

At o° C. water freezes if further heat is withdrawn, and by 
everything we leam from other solidifications we would have a 
right to expect a sharp increase of density. We would be wrong! 
There is a sharp decrease in density. 

Whereas water at o° C. has, as 1 said, a density of 0.9999 grams 
per cubic centimeter, it freezes into ice at o° C. with a density of 
only about 0.92 grams per cubic centimeter. 

If a cubic centimeter of ice is completely immersed in water, 
with both at a temperature of o° C, then the weight of the ice is 
— 0.08 grams and there is, so to speak, a negative gravitational ef- 
fect upon it. It therefore rises to the surface of the water. The 
rise continues till only enough of it is submerged to displace its 
own weight (as measured in air) of the denser, liquid water. Since 
a cubic centimeter of ice at o° C. weighs 0.92 grams and it takes 
only 0.92 cubic centimeters of water at o° C. to weigh 0.92 

* 3.98° C, to be more accurate. 



grams, it turns out that when the ice is floating, 92 per cent of its 
substance is below water and 8 per cent is above. 

What we would ordinarily expect, judging from almost all other 
solids immersed in their own liquid form, is that 100 per cent of 
the ice would be submerged and o per cent exposed. It follows, 
then, as I said earlier, that the surprising thing is not that so much 
of an iceberg is invisible, but that so much of it (or, indeed, any of 
it at all) is exposed. 

Well, why is that? 

Let's begin with ice. In ordinary ice, each water molecule has 
four other molecules surrounding it with great precision of orien- 
tation. The hydrogen atom of each water molecule is pointed in 
the direction of the oxygen atom of a neighbor and this orienta- 
tion is maintained through the small electrostatic attraction in- 
volved in the hydrogen bond (as described in the previous 

The hydrogen bond is weak and does not suffice to draw the 
molecules very close together. The molecules remain unusually far 
apart, therefore, and if a scale model is built of the molecular 
structure of ice, it is seen that there are enough spaces between 
the molecules to make up a very finely ordered array of "holes." 
Nothing visible, you understand, for the holes are only about an 
atom or so in diameter. 

Still, this makes ice less dense than it would be if there were 
a closer array of molecules. 

As the temperature of the ice rises, its molecules vibrate and 
move still farther apart, so that its density falls, reaching a 
minimum of the aforementioned 0.92 grams per cubic centimeter 
at o° C. At that temperature of o° C, however, the molecular 
vibration has reached the point where it just balances the attrac- 
tive forces between the molecules. If further heat is added, the 
molecules can break free and can begin to slip and slide past each 
other. In doing so, however, some of them fall into the holes. 

As ice melts, then, the tendency to decrease the density 
through increased vibrational energy is countered by the dis- 



appearance of the holes, and more than countered. For that rea- 
son, liquid water is 8 per cent denser at o° C. than solid water is. 

Even in water at o° C, however, the loose molecular arrange- 
ment in ice hasn't utterly vanished. As the temperature rises still 
higher, there is still a slow disappearance of the last few lingering 
holes and it is not till a temperature of 4° C. is reached that so 
few of them are left that they can no longer exert a dominating 
effect on the density change. At temperatures higher than 4° C, 
the energy of molecular vibration increases and density decreases 
steadily as it "ought" to do. 

The importance of this density anomaly in water simply can't 
be exaggerated. Consider what happens to a moderately sized lake, 
for instance, during a cold winter. 

The temperature of the water gradually drops from its mild 
warmth of the summer. Naturally, it is the water at the surface 
that cools first, becomes denser, and sinks, forcing up the warmer 
water at the bottom, so that it can, in turn, cool and sink. In this 
way the entire body of water cools, and would cool all the way to 
o° C. if the density continued to increase steadily as temperature 

As it is, though, when a temperature of 4° C. is reached, a fur- 
ther cooling of the surface water makes it slightly less dense! It 
does not sink, but floats on the warmer water below. The surface 
water drops in temperature all the way to o° C, but heat leaves the 
lower depths only slowly and those depths remain somewhat 
warmer than o° C. 

It is the water at the surface, then, that experiences freezing, 
and the ice, being less dense than water, remains floating. If the 
cold weather continues long enough, the entire layer of surface 
water freezes and forms a solid coating of ice that may become 
thick and strong indeed (to the satisfaction of ice skaters). 

But ice is a good insulator of heat, and the thicker it is, the 
more effective an insulator it is. As it thickens, the deeper layers 
of water (still liquid) lose heat through the ice to the air above 
more and more slowly; and more and more slowly does the ice 


layer thicken further. In short, in any winter that is likely to occur 
on Earth, a sizable lake will never freeze solidly all the way to the 
bottom. This means that life-forms in it can survive through the 

What's more, when the warm weather returns, it is the surface 
ice that receives the brunt of the Sun's heat. It melts and the 
liquid water beneath is at once exposed, so that the lake quickly 
becomes liquid throughout once more. 

What would happen, though, if water were like other sub- 
stances. In cooling, there would be a continual sinking of cooler 
water all the way down to o° C, so that the entire body of the lake 
would be at that temperature eventually. It would have a tend- 
ency to freeze at every point, and any ice that formed near the 
surface of the lake would sink at once if there were still liquid 
below it. A winter that under present circumstances would only 
suffice to form a thick scum of ice on a lake would be enough to 
freeze that same lake solid, top to bottom, if water were like other 

Then, when warm weather came, the surface of the frozen lake 
would melt, but the water that formed would insulate the deeper 
layers of ice from the Sun's heat. The thicker the layer of liquid 
water, the more slowly the Sun's heat would penetrate to the ice 
below and the more slowly would the deeper ice melt. Through 
an ordinary summer such as we experience on Earth, a solidly 
frozen lake would never melt all through. Most of it would remain 
permanently frozen. 

The same would hold true for rivers and for the polar oceans. 
Indeed, if water were suddenly to change its density character- 
istics, each winter would see further ice form and sink to the 
ocean abyss to remain permanently frozen thereafter. Eventually, 
all of Earth would be a mass of ice-bound land, with a thin layer 
of water on the surface of the tropic ocean. 

Even though such an Earth would be at the distance from the 
Sun it now is, and would receive the amount of solar energy it 
now does, it would be a frigid world and life as we know it would 
not have formed. It follows then that life depends on the hydrogen 



bond not merely for the reasons I outlined in the last chapter, but 
because of the loose structure it gives ice. 

There's another way to break down the holes in ice besides 
raising the temperature. Why not simply squeeze the ice to- 
gether under pressure? To be sure, it takes enormous pressures to 
squeeze out the holes to the point where ice is as dense as water. 
(When water is allowed to fill a sealed container tightly and then 
made to freeze, it exerts an outward pressure equal to the pressure 
it would take to compress ice to the density of water — and the 
container breaks.) 

Still, high pressures can be produced in the laboratory. About 
1900, a German physicist, Gustav Tammann, began to make use 
of such high pressures, and beginning in 1912, an American phys- 
icist, Percy W. Bridgman, carried the matter much further. 

In this way, it was found that there were many forms of ice. 

In any solid there is an orderly arrangement of molecules and 
there is always the possibility of a variety of different arrange- 
ments under different conditions. Some anangements are more 
compact than others and these would be favored by high pressures 
and low temperatures. 

Thus, under ordinary temperatures and pressures, ordinary ice 
(which we can call Ice 1) is the only variety that can exist. As the 
pressure is increased, however, two other forms are found. Ice 
11 at temperatures below — 35° C, and Ice ni at temperatures be- 
tween —35° C. and —20° C. 

If the pressure is raised still further. Ice v is formed. (There is 
no Ice rv; it was reported but proved to be a case of mistaken 
observation and was dropped; but not before Ice v had been re- 

If the pressure is raised still further. Ice vi and Ice VII are 
formed. Whereas all other forms of ice exist only at o° C. and 
below. Ice vi and Ice VII can exist at temperatures above o° C, 
though only at enormous pressures. 

In fact, at a pressure of 20,000 kilograms per square centimeter 
(one and a half million times the pressure of the atmosphere), 


Ice vri will exist at temperatures above ioo° C, the boiling point of 
liquid water under ordinary conditions. 

All these high-pressure forms of ice are denser than liquid wa- 
ter, as you would expect, for the holes have been squeezed out of 
them. Indeed, of all known forms of ice, only Ice i, the ordinary 
variety, is less dense than liquid water. 

It would follow that if any of the forms of ice other than Ice i 
could form in the oceans, they would sink to the bottom and 
gradually accumulate. 

In one of his excellent novels, Kurt Vonnegut hypothesized a 
mythical "Ice rx," which could exist at the ocean bottoms and which 
would form spontaneously if only some small quantity existed as a 
"seed." The hero had such a small piece and, of course, it got into 
the ocean to bring about the final catastrophe. 

Is there really a chance of that? No. Any form of ice but Ice i 
can only exist at enormous pressures. Even the least high-pressure 
ices (Ice n and Ice in) can exist only at pressures more than two 
thousand times that of the atmosphere. If such pressures could 
be attained at the bottom of the ocean (they can't), a further 
requirement would be that the temperature be well below — 20° C. 
(It isn't.) 

You can see, further, that no form of ice other than Ice 1 could 
exist in someone's pocket. If any other ice were formed and the 
high pressure required to form it were removed, the ice would 
instantly expand to Ice 1 with explosive violence. 

That still leaves one thing to discuss. Though solid forms of a 
substance can (and often do) exist in a variety of crystalline 
forms, liquid and gaseous forms do not. In liquids and gases there 
is not, generally speaking, any orderly array of molecules, and one 
does not find varieties of disorder. 

But in 1965, a Soviet scientist, B. V. Deryagin, studied liquid 
water in very thin capillary tubes and found some of it to possess 
most unusual properties. For one thing, its density was 1.4 times 
that to be expected of ordinary water. Its boiling point was ex- 
traordinarily high and it could be heated up to 500° C. before 



ceasing to be liquid. It could be cooled down to — 40° C. before 
turning into a glassy solid. 

The report was largely disbelieved in the West, where there is 
almost automatically skepticism toward any unusual finding that 
emerges outside the charmed circle of nations prominent in 
nineteenth-century science. 

However, when Americans repeated Deryagin's work, they 
found, much to their own surprise, that they got the same results 
and could even see droplets of the anomalous form of liquid 
water — droplets so small they could be made out only under a 

What was behind this? 

Water molecules, while slipping and sliding around each other, 
do tend to take up the hydrogen bond orientation, as in ice. This 
happens over very small volumes and for very brief periods, but 
it is enough to make liquid water behave as though it consisted of 
submicroscopic particles of ice that form and unform with super- 

The "ice" never forms over a volume large enough and for a 
time long enough to make the holes significant and cause water 
to be as un-dense as ice, but it does keep the water molecules far 
enough apart to allow hydrogen bonds to form and unform. 
Liquid water is therefore less dense than it might be. 

Suppose, though, that pressure is placed on water in such a way 
that molecules are forced closer together while in the hydrogen 
bond orientation. With neighboring molecules unusually close, 
the hydrogen bond would be much stronger than ordinary and 
would, indeed, approach an ordinary chemical bond in strength. 
Molecule after molecule would fall into place and, thanks to the 
unusually strong hydrogen bond attractions, they would make up 
a kind of giant molecule built up out of the small water-molecule 

When small units build up a giant molecule in this fashion, the 
small units are said to "polymerize" and the giant molecule is a 
"polymer." The new form of water was therefore spoken of as 
"polymerized water" or, for short, "poly water." 

102 THE problem of oceans 

In polywater, the molecules are in orderly array, something as 
in ice, but in much more compact fashion, and certainly without 
the holes. Not only does this compact airay of water molecules 
produce a substance considerably more dense than ice, but con- 
siderably more dense than ordinary liquid water as well. 

What's more, because the molecules are held more tightly to- 
gether, it takes a much higher temperature than ioo° C. to tear' 
them apart and make poly water boil. It also takes a much lower 
temperature than o° C. to force the molecules apart into the 
less compact array of ordinary ice. Other unusual properties of 
polywater are also easily explained on the basis of the compact 
array of molecules. 

Apparently, polywater does not form under ordinary increases 
in pressure, but does form in the constricted volume of tiny 
capillary tubes. Biologists at once began to wonder whether within 
the constricted volume of tissue cells, polywater also formed; and 
whether some of the properties of life could not be most easily 
explained in terms of polywater. 

I wish I could end the matter here, with this glamorous dis- 
covery and the still more glamorous speculation, but I can't. The 
trouble is that many chemists remain skeptical of the whole busi- 

It is possible, after all, that investigator's have been misled by 
the chance of solution of the glass horn the tubes in which the 
polywater was being studied. If it were not pure liquid water they 
were studying but tiny volumes of glass solution, all bets were 

Indeed, one chemist recently prepared a solution of silicic acid 
(something which could form when water is in contact with glass) 
and reported it to possess the very properties of polywater. 

So it may be that polywater is a false alarm, after all. 

C — The Problem of Numbers and Lines 


Not long ago I got a letter from a young amateur mathematician 
which offered me a proof that the number of primes was infinite 
and asked, first, if the proof were valid, and, second, if it had ever 
been worked out before. 

I answered that first, the proof was a valid and elegant one but 
second, that Euclid had worked out the same proof, just about 
word for word, in 300 B.C. 

Alas, alas, this is the fate of almost every single one of us ama- 
teur mathematicians almost every single time. Anything we work 
out that is true is not new; anything we work out that is new is not 
true. — And yet, if we work out what is true, from a standing start, 
without ever having had it worked out for us, then I maintain it 
to be a feat of note. It may not advance mathematics, but it is a 
triumph of the intellect just the same. 

I told my young correspondent this and now I would like to tell 
you about the proof and about a few other things. 

First, what is a prime or, more correctly, a "prime number"? A 
prime is any number that cannot be expressed as the product of 
two numbers, each smaller than itself. Thus, since 15 = 3X5, 15 
is not a prime. On the other hand, 13 cannot be expressed as a 
product of smaller numbers and is therefore a prime. Of course, 
13 = 13X1, but 13 is not smaller than 13, so that this multiplica- 
tion does not count. Any number can be expressed as itself multi- 
plied by 1, whether it is prime or not (15 = 15 X 1, for instance), 
and this sort of business is no distinction. 

Another way of putting it is that a prime number cannot be di- 
vided evenly ("has no factors") other than by itself and by 1. Thus 
15 can be divided evenly by either 3 or 5, in addition to being di- 


visible by X5 and 1; but 13 can be divided only by X3 and 1. So 
again, 15 is not a prime and X3 is. 

Well then, what numbers are prime? Alas, that is not an easy 
question to answer. There is no general way of telling a prime 
number just by looking at it. 

There are certain rules for telling if a particular number is not 
a prime, but that is not the same thing. For instance, 287,444,409,- 
786 is not a prime. I can tell that at a glance. What's more, 
287,444,409,785 is not a prime, either, and I can tell that at a 
glance, too. But is 287,444,409,787 a prime? All I can tell at a 
glance is that it may be a prime; but also it may not. There is no 
way I can tell for certain unless I look it up in a table — assum- 
ing that I have a table that gives me all the prime numbers up to a 
trillion. If I don't have such a table, and I don't, I have to sit 
down with pen and paper and try to find a factor. 

Is there any systematic way of finding all the primes up to some 
finite limit. Yes, indeed, there is. Write down all the numbers from 
1 to xoo. (I'd do it for you here, but I'd waste space, and it will 
be good exercise for you if you do it.) 

The first number is x but that is not a prime by definition. The 
reason for that is that in multiplication — which is the way we have 
of distinguishing primes from non-primes — the number x has the 
unique property of not changing a product. Thus 15 could be writ- 
ten as 5 X 3, or as 5 X 3 X x, or, indeed, as 5X3X1X1X1X1 
. . . and so on forever. By simply agreeing to eliminate x 
from the list of primes, we eliminate the possibility of a tail of x's, 
and get rid of some nasty complications in the theoretical work on 
primes. No other number acts like x in this respect and no other 
number requires special treatment. 

We next come to 2, which is a prime since it has no factors 
other than itself and x. Let's eliminate every number in our list 
that can be divided by 2 (and is therefore non-prime) and to do 
that we need only cross out every second number after 2. This 
means we cross out 4, 6, 8, xo, and so on, up to and including 
100. You can check for yourself that these numbers are not prime, 
since 4 = 2X2; 6 = 2X3; 8 = 2X4, and so on. 


We look at our list of numbers and find that the smallest num- 
ber not crossed out is 3. This is a prime since 3 has no factor's other 
than itself and 1. So we begin with 3 and cross out eveiy third 
number after it: 6, 9, 12, 15, and so on, up to and including 99. 
Some of the numbers, 6 and 12, for instance, were already 
crossed out when we were dealing with 2, but that's all right; cross 
them out again. The numbers now crossed out are all divisible by 
3 and are therefore not prime: 6 = 3X2;9 = 3X3, and so on. 

The next number not crossed out is 5, and you cross out eveiy 
fifth number after it. Then 7, and you cross out every seventh 
number after it. Then 11, then 13, and so on. By the time you 
reach 47 and proceed to cross out the 47th number after it (94), 
you find you have crossed out every number below 100 that you 
can. The next available number is 53, but if you tiy to cross out 
the 53d number after it, that is 106, which is beyond the end of 
the list. 

So you have left the following numbers under 100 which are 
not crossed out: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 
53, 59, 61, 67, 71, 73, 79, 83, 89, 97. 

These are the twenty-five prime numbers below 100. If you 
memorize them you will be able to tell at a glance whether any 
particular' number under 100 is a prime or not, just by knowing 
whether it is or is not on the list I just gave you. 

Is there a simple connection among all these numbers, some 
formula that will give only the primes up to 100 and no other 
numbers? Even if you could work out such a formula, it wouldn't 
help you, for it would break down as we proceed above 100, for 
after all, we can, if we want to, continue to use the same sys- 
tem of stopping at each uncrossed number and counting off every 
one that is its own number after it. We would then find out that 
above 100, there are prime numbers such as 101, 103, 107, 109, 
113, 127, and so on. 

If we had written all the numbers up to 1,000,000,000,000, we 
would eventually have worked out all the primes up to that point 
and we would have determined, mechanically and without flaw 


(provided we make no mistake in counting), whether the number 
I gave you previously, 287,444,409,787, is or is not a prime. 

This perfect system for finding all the prime numbers up to any 
finite number, however large, is called "the sieve of Eratosthenes," 
because the Greek scholar Eratosthenes first used it somewhere 
about 230 B.C* 

There is one trouble with the sieve of Eratosthenes and that 
is that it takes an unconscionable length of time. Working it out 
through 100 isn't bad, but time yourself working it through 1,000 
or through 10,000 and you'll agree that it soon piles up prohibi- 

But wait. After all, you keep piling up more and more prime 
numbers and each one sieves out some of all the still higher num- 
bers remaining. This means that a larger and larger percentage of 
those still higher numbers is crossed out, doesn't it? 

Yes, it does. There are twenty-five primes under 100, as I just 
pointed out, but only twenty-one primes between 100 and 200, 
and sixteen primes between 200 and 300. This dwindling is an ir- 
regular thing and sometimes the number jumps, but on the whole, 
the percentage of primes does dwindle — there are only eleven 
primes between 1,300 and 1,400. 

Well, then, do the primes ever come to a complete halt? 

Put it another way. As one goes up the line of numbers, there 
are longer and longer intervals, on the average, between primes. 
That is, there are longer and longer lists of successive non-primes. 
The longest successive stretch of non-primes under 30 is five: 24, 
25, 26, 27, and 28. There are seven successive non-primes be- 
tween 89 and 97; thirteen successive non-primes between 113 and 
127, and so on. If you go high enough, you will find a hundred suc- 
cessive non-primes, a thousand successive non-primes, ten thou- 
sand successive non-primes, and so on. 

* When Frederik Pohl (the well-known science fiction writer and editor) 
was young, he worked out the sieve of Eratosthenes all by himself and 
was most chagrined to find out he had been anticipated. But one needs 
no more evidence than that to demonstrate Fred's brightness. Working it 
out independently (simple though it seems after it is explained) was more 
than I was ever able to do. 



You can find (in theory) any number of non-primes in succes- 
sion no matter how high a number you name, if you proceed along 
the list of numbers long enough. But, and this is a big "but," is 
there ever a time when the number of non-primes in succession is 
infinite? If so, then after a certain point in the list of numbers, 
all the remaining numbers will be non-prime. The number mark- 
ing that "certain point" would be the largest prime number pos- 

What we are asking now, then, is whether the number of primes 
is infinite or whether there is, instead, some one prime that is the 
largest of all, with nothing prime beyond it. 

Your first thought might be to work out the siejfe of Eratos- 
thenes till you reach a number beyond which you can see that 
everything higher is crossed out. That, however, is impossible. No 
matter how high you go, and how long a vista thereafter seems to 
be non-prime, you can never possibly tell whether there is or is 
not another prime somewhere (perhaps a trillion numbers fur- 
ther) up ahead. 

No, you must use logical deduction instead. 

Let's consider a non-prime number that is a product of prime 
numbers: say, 57 = 19 X 3. Now let's add 1 to 57 and make it 58. 
The number 58 is not divisible by 3, since if you try the division 
you get 19 with a remainder of 1; nor is it divisible by 19, for that 
will give you 3 with a remainder of 1. This is not to say that 58 is 
not divisible by any number at all, for it is divisible by 2 and by 
29 (58 = 2 X 29). 

You can see, however, that any number that is the product of 
two or more smaller numbers, is no longer divisible by any of those 
numbers if its value is increased by 1. To put it in symbols: 

IfN = P x Q x R . . . . , then N -I- 1 is not divisible by either 
P or Q or R or any other factor of N. 

Well, then, suppose you begin with the smallest prime, 2, and 
consider the product of all the successive primes up to some point. 
Begin with the two smallest primes: 2 X 3 = 6. If you add 1 to the 
product you get 7, which cannot be divided by either 2 or 3. As a 
matter of fact, 7 is a prime number. You go next to (2X3X5)4-1 

= 31 and that's a prime. Then (2X3X5X7)4-1=211, and 
(2 X 3 X 5X 7X 11) + 1 =2311, and both 211 and 2311 are 

If we then try (2X3X5X7X11X13)+!, we get 30,031. 
That, actually, is not a prime number. However, neither 2, 3, 5, j, 
11, nor 13 (which represent all the primes up to 13) are among its 
factors, so any primes that must be multiplied to make 30,031, 
must be higher than 13. And, indeed 30,031 = 59 X 509. 

We can say, as a general rule, that 1 plus the product of any 
number of successive primes, beginning with 2 and ending with P, 
is either a prime itself and is therefore certainly higher than P, or 
is a product of prime numbers all of which are higher than P. 
And since this is true for any value of P, there can be no highest 
prime, since a mechanism exists for finding a still higher one, no 
matter how high P is. And that, in turn, means that the number of 
primes is infinite. 

This, in essence, is the proof Euclid presented, and it is the 
proof my young correspondent worked out independently. 

The next problem is this: Granted that the number of primes 
is infinite, is there any formula that has as its solution all the 
primes and none of the non-primes, so that we can say: Any num- 
ber that is a solution of this formula is a prime; all others are 
not? You see, to determine whether 287,444,409,787 is a prime or 
not by the sieve of Eratosthenes, which will surely tell you, you 
must work your way up through all the lower numbers. You can't 
skip. A "prime-formula" will enable you to crank in 287,444,409,- 
787 directly and tell you whether it is prime or not. 

Alas, there is no such formula, and it is not likely that any can 
ever be found (although I am not sure that it has been proven 
that none can be found). The order of primes along the list of 
numbers is utterly irregular and no mathematician has ever been 
able to work out any order, however complicated, which would 
make a "prime-formula," however complicated, possible. 

Let's lower our sights then. Is it possible to work up some useful 
formula that will give us not every prime, but at least only primes? 



We could then grind out an infinite series of known primes by 
turning a formula-crank, even though we know we are skipping an 
infinite quantity of other primes. 

Again, no (except for some specialized non-useful cases). No 
matter how we try to find a useful system that will yield primes 
only, non-primes will always sneak in. For instance, you might 
think that adding 1 to the product of successive primes beginning 
with 2 might yield only prime numbers. The numbers I got this 
way a little earlier in the article were 7, 31, 211, 2311 — all primes! 
But then, the next in the series was 30,031, and that was not a 

Formulas have been worked out in which the value n was sub- 
stituted by the numbers 1, 2, 3, and so on, with prime values ob- 
tained for every value up to n = 40. And then for n = 41, a non- 
prime will pop out. 

So let's lower our sights again. Is there any system that will allow 
us to crank out only non-primes? Non-primes may not be inter- 
esting but at least we can get rid of them and study a group of re- 
maining numbers that will be denser in primes. 

Yes! At last we have something to which the answer is, yes! In 
working out the sieve of Eratosthenes, for instance, perhaps you 
noticed that in crossing out every second number after 2, you 
crossed out only numbers that ended with 2, 4, 6, 8, and o, and 
that you crossed out every number that ended with 2, 4, 6, 8, 0. 
This means that any number, no matter how long and formi- 
dable, even if it has a trillion digits, is not a prime if the last digit is 
2, 4, 6, 8, or o; if it's an "even number," in other words. 

Since exactly half of all the numbers in any finite successive list 
end in these digits, that means that all primes (except for 2 itself, 
of course) must exist in the other half — the odd numbers. 

Then again, when you begin with 5 and cross out every fifth 
number, you cross out only numbers that end with 5 and o, and 
every number that ends with 5 and 0. Numbers ending with o are 
already taken care of, but now we can eliminate any number from 
the list of possible primes if the last digit is 5 (except for 5 itself, 
of course). 


This means we need look for primes (other than 2 and 5) only 
in those numbers that end in the digits 1, 3, 7, or 9. This means 
that in any successive list of numbers we can eliminate 60 per cent 
and look for primes only in the remaining 40 per cent. 

Of course, if we take into account not a finite successive list of 
numbers (say from 1 to 1,000,000,000,000) but all numbers, the 
40 per cent that may contain primes is still infinite and still con- 
tains an infinite number of primes — and an infinite number of 
non-primes, too. Restricting the places in which we look for primes 
doesn't help us in the ultimate problem of finding all the primes 
by some mechanical method easier than the sieve of Eratosthenes 
but at least it clears away some of the underbrush. 

Of course, there are other possible eliminations. Any number, 
no matter how long and complicated, whose digits add up to a 
sum divisible by 3 is itself divisible by 3 and is not a prime. How- 
ever, digit adding is tedious, so let's restrict ourselves to just look- 
ing at the last digit. The trick of looking at the last digit is the 
only elimination device that is simple enough to be pleasing. Is 
there anything we can do to improve the situation that exists? 

To answer that, let us ask what the magic is of 2 and 5 that en- 
ables them to make their mark on the final digit. The answer is 
easy. Our number system is based on 10 and 10 = 2 X 5. What we 
have to do is find a number that is the product of two separate 
primes that is smaller than 10. Maybe we can then crowd the 
"magic" into a smaller area. 

Only one number smaller than 10 will do and that is 6 = 2 X 3. 

All numbers are either multiples of 6 or, on being divided by 6, 
leave remainders that are equal to 1, 2, 3, 4, or 5. There are no 
other possibilities. This means that any number is of the class 6rz, 
6n + 1, 6n + 2, 6n + 3, 6n + 4, or 6n + 5. Of these, any number 
of the form 6n cannot be a prime since it is divisible by both 2 and 3 
(6n = 2 X 371 = 3 X in). Any number of the form 6n + 2 or 6n + 4 
is divisible by 2 and any number of the form 6n + 3 is divisible by 3. 

That means that all primes (except 2 and 3) must be of the 
forms 6n + 1 or 6n + 5. Since 6n + 5 is equivalent to 6n — 1, we 


might say that all prime numbers are either one more or one less 
than a multiple of 6. 

Suppose, then, we make a list of multiples of 6: 6, 12, 18, 24, 
30, 36, 42, 48 , 54, 60, 66, 72, 78, 84, 90, 96, 102 .. . 

With that as a guide, we could next make a double list of all 
numbers one less than these multiples, and one more, with bold 
face for those numbers which are prime: 

5,1 1,17,23,29, 35,41,47,53,59,65,71,77, 8 3,89,95,101.. . 
7,13,19,25,31,37,43,49, 55, 61,67,73,79,85,91,97,103 .. . 

As you see, the numbers in the list occur in pairs of which one 
is 2 more than the other (with a multiple of 6 in between). You 
might think, after looking at the list above, that at least one of 
each pair must be a prime and that that imposes some kind of ad- 
ditional order on the primes. That is not so, unfortunately. At 
least one of each pair is a prime as far as we've gone, but if you go 
further, you will find that in the pair 119, 121, neither one is a 
prime. The number 119, which is 6 X 20 — 1, is equal to 7 X 17 
and 121, which is 6 X 20 + 1, is equal to 11 X 11. The higher 
up you go the more common the non-prime pairs get. 

Sometimes only the upper and smaller number of the pair is a 
prime, as in 23 and 25; sometimes only the lower and higher 
number, as in 35 and 37. In the end, both upper and lower lists get 
an equal share but in an absolutely irregular fashion. 

There are also occasions when both numbers of the pair are 
prime, as in 5 and 7, 11 and 13, and 101 and 103. Such pairs are 
called "prime twins" and they can be found as far as the list of 
numbers has been investigated for primes. The density of their 
occurrence diminishes as the numbers grow larger, just as does the 
density of the primes themselves. It would seem, however, that 
the density of prime twins never falls to zero and that the number 
of prime twins is infinite. That, however, has never been proved. 

If we consider the numbers of the form 6n + 1 and 6n — 1 only, 
we find they contain every single prime in existence (except 2 and 
3) yet make up only one third of all the numbers in any finite suc- 
cessive list. Is there any way we can translate this into the final- 
digit business? 




The answer is yes!!!! And I use those exclamation points be- 
cause I come here to something that I am sure has been well 
known to mathematicians for at least two centuries, but which I 
have never seen mentioned in any book I have read. I have worked 
this out independently! 

All you have to do is use a six-based system, in which our ordi- 
nary numbers look as follows: 

10-based: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 . . . 

6-based: 1,2,3,4,5,10,11,12,13,14,15,20,21,22,23,24,25. . . 
(There is no space here to go into details on other-based number 
systems, but see "One, Ten, Buckle My Shoe," reprinted in Adding 
a Dimension, Doubleday, 1964.) 

In the 6-based system, only numbers ending in the digits one 
and five could possibly be prime. In the 6-based system we would 
know at once that 14313234442, 14313234443, 14313234444, and 
14313234440 were not prime, just by looking at the last digit. On 
the other hand, 14313234441 and 14313234445 might be prime 
(and, unfortunately, might not). 

The point is that in a 6-based system you could instantly elimi- 
nate two thirds of the numbers in any finite successive list of num- 
bers just by looking at the final digit, leaving one third to contain 
all the primes (except 2 and 3). This is better than we can do 
in the 10-based system, where we eliminate three fifths and leave 
two fifths. 

But what if we use a number as base that does not have two dif- 
ferent prime factors as do 6 and 10, but three different prime 
factors? The smallest number which qualifies is 30 = 2 X 3 X 5. 

If we use 30 as base, consider that all numbers are of the form 
3on, 3on + 1, 3071 + 2, 300 4- 3 ... all the way up to 3on + 29. 
Of these, numbers of the form 30/1, 3071 + 2, 30/1 + 4, and so on, 
are divisible by 2 and are therefore non-prime; numbers of the 
fomr 30/1 + 3, 3011 + 9, 3011 +15, and so on are divisible by 3 
and are therefore non-prime; numbers of the form 3071 + 5 and 
3on 4- 25 are divisible by 5 and are therefore non-prime. In the 
end, the only numbers that cannot be divided by 2, 3, or 5 (ex- 
cept for 2, 3, and 5 themselves) and therefore may be primes, are 


numbers of the classes -}on + 1, -ym + 7, 3071 + 1 1, 3071 + 1 3, 
3on + 17, 3on + 19, 3071 + 23, and $on + 29. 

This sounds like a large number of classes to contain primes, but 
in the 30-based system there are thirty different digits, one repre- 
senting every number from o to 29 inclusive. And in a 30-based 
system, numbers ending in twenty-two of these thirty digits are 
non-prime on the face of it. Only those ending in the eight digits 
equivalent to our ten-based numbers 1, 7, 11, 13, 17, 19, 23, and 
29 may be primes. 

In the 30-based system, then, we eliminate eleven fifteenths, or 
73% per cent, of any finite successive list of numbers and crowd 
all the primes (except 2, 3 and 5) into the 26% per cent remain- 

Of course, you can go still higher. You can use a number system 
based on 210 (since 210 = 2X3X5X7) or 2310 (since 
2310 = 2 X 3 X 5 X7X a) or still higher, going up the scale 
of multiplied primes as far as you care to go. In each case, you have 
to leave out of account all the primes that are factors of the num- 
ber base, but will find all other primes crowded into a smaller and 
smaller fraction of any finite successive list of numbers. 

Here's the way it works as far as I've gone: 

Number Base 

% eliminated 

% remaining 







2X3X5 = 30 



2X3X5X7 = 210 



I refuse to go higher. You can work it out for 2310 or for any 
still higher number base yourself.* 

Now mind you, the larger you make the number on which you 
base your number system, the more inconvenient it is to handle 
that system in practice, regardless of how beautiful it may be in 

* Since this chapter first appeared, knowledgeable readers have sent me 
formulas to use in such calculations. If I had known them I would have had a 
lot less trouble. 


theory. It is perfectly easy to understand the system for writing 
and handling numbers in a 30-based system, but to try to do so 
in actual manipulations on paper is a one-way ticket to the booby 
hatch — at least if your mind is no nimbler than mine. 

The gain in prime-concentration in passing to a 30-based sys- 
tem (and I won't even talk about a 210-based system or anything 
higher) is simply not worth the tremendous loss in manipulability. 

Let us therefore stick with the 6-based system, which is not only 
more efficient as a prime-concentrator than our ordinary 10-based 
system is, but is actually easier to manipulate once you are used 
to it. 

Or we can put it another way. It is the 6-based system which is, 
in this respect at least, of prime quality.* 

* Let there be no groaning in the galleryl 


Some of my articles stir up more reader comment than others, 
and one of the most effective in this respect was one I once wrote 
in which I listed those who, in my opinion, were scientists of the 
first magnitude and concluded by working up a personal list of 
the ten greatest scientists of all time. 

Naturally, I received letters arguing for the omission of one or 
more of my ten best in favor of one or more others, and I still 
get them, even now, seven and a half years after the article was 

Usually, I reply by explaining that estimates as to the ten great- 
est scientists (always excepting the case of Isaac Newton, concern- 
ing whom there can be no reasonable disagreement) are largely a 
subjective matter and cannot really be argued out. 

Recently, I received a letter from a reader who argued that 
Archimedes, one of my ten, ought to be replaced by Euclid, who 
was not one of my ten. I replied in my usual placating manner, 
but then went on to say that Euclid was "merely a systematizer" 
while Archimedes had made very important advances in physics 
and mathematics. 

But later my conscience grew active. I still adhered to my own 
opinion of Archimedes taking pride of place over Euclid, but the 
phrase "merely a systematizer" bothered me. There is nothing nec- 
essarily "mere" about being a systematizer.* 

For three centuries before Euclid (who flourished about 300 
B.C.) Greek geometers had labored at proving one geometric 
theorem or another and a great many had been worked out. 

What Euclid did was to make a system out of it all. He began 

* Sometimes there is. In all my non-fiction writings I am "merely" a 
systematizer. — Just in case you think I'm never modest. 


with certain definitions and assumptions and then used them to 
prove a few theorems. Using those definitions and assumptions 
plus the few theorems he had already proved, he proved a few ad- 
ditional theorems and so on, and so on. 

He was the first, as far as we know, to build an elaborate mathe- 
matical system based on the explicit attitude that it was useless to 
try to prove everything; that it was essential to make a beginning 
with some things that could not be proved but that could be ac- 
cepted without proof because they satisfied intuition. Such in- 
tuitive assumptions, without proof, are called "axioms." 

This was in itself a great intellectual advance, but Euclid did 
something more. He picked good axioms. 

To see what this means, consider that you would want your list 
of axioms to be complete, that is, they should suffice to prove all 
the theorems that are useful in the particular field of knowledge 
being studied. On the other hand they shouldn't be redundant. 
You don't want to be able to prove all those theorems even after 
you have omitted one or more of your axioms from the list; or to 
be able to prove one or more of your axioms by the use of the re- 
maining axioms. Finally, your axioms must be consistent. That 
is, you do not want to use some axioms to prove that something is 
so and then use other axioms to prove the same thing to be not so. 

For two thousand years, Euclid's axiomatic system stood the 
test. No one ever found it necessary to add another axiom, and 
no one was ever able to eliminate one or to change it substantially 
— which is a pretty good testimony to Euclid's judgment. 

By the end of the nineteenth century, however, when notions 
of mathematical rigor had hardened, it was realized that there 
were many tacit assumptions in the Euclidean system; that is, 
assumptions that Euclid made without specifically saying that he 
had made them, and that all his readers also made, apparently 
without specifically saying so to themselves. 

For instance, among his early theorems are several that demon- 
strate two triangles to the congruent (equal in both shape and 
size) by a course of proof that asks people to imagine that one tri- 
angle is moved in space so that it is superimposed on the other. 



— That, however, presupposes that a geometrical figure doesn't 
change in shape and size when it moves. Of course it doesn't, you 
say. Well, you assume it doesn't and I assume it doesn't and Euclid 
assumed it doesn't — but Euclid never said he assumed it. 

Again, Euclid assumed that a straight line could extend infi- 
nitely in both directions — but never said he was making that as- 

Furthermore, he never considered such important basic proper- 
ties as the order of points in a line, and some of his basic defini- 
tions were inadequate — 

But never mind. In the last century. Euclidean geometry has 
been placed on a basis of the utmost rigor and while that meant 
the system of axioms and definitions was altered, Euclid's geom- 
etry remained the same. It just meant that Euclid's axioms and 
definitions, plus his unexpressed assumptions, were adequate to 
the job. 

Let's consider Euclid's axioms now. There were ten of them and 
he divided them into two groups of five. One group of five was 
called "common notions" because they were common to all sci- 

1) Things which are equal to the same thing are also equal to 
one another. 

2) If equals are added to equals, the sums are equal. 

3) If equals are subtracted from equals, the remainders are 

4) Things which coincide with one another are equal to one 

5) The whole is greater than the part. 

These "common notions" seem so common, indeed so obvious, 
so immediately acceptable by intuition, so incapable of contradic- 
tion, that they seem to represent absolute truth. They seem some- 
thing a person could seize upon as soon as he had evolved the light 
of reason. Without ever sensing the universe in any way, but liv- 
ing only in the luminous darkness of his own mind, he would see 

that things equal to the same thing are equal to one another and 
all the rest. 

He might then, using Euclid's axioms, work out all the theorems 
of geometiy and, therefore, the basic properties of the universe 
from first principles, without having observed anything. 

The Greeks were so fascinated with this notion that all mathe- 
matical knowledge comes from within that they lost one impor- 
tant urge that might have led to the development of experimental 
science. There were experimenters among the Greeks, notably 
Ctesibius and Hero, but their work was looked upon by the Greek 
scholars as a kind of artisanship rather than as science. 

In one of Plato's dialogues, Socrates as ks a slave certain ques- 
tions about a geometric diagram and has him answer and prove a 
theorem in doing so. This was Socrates' method of showing that 
even an utterly uneducated man could draw truth from out of 
himself. Nevertheless, it took an extremely sophisticated man, 
Socrates, to ask the questions, and the slave was by no means un- 
educated, for merely by having been alive and perceptive for years, 
he had learned to make many assumptions by observation and 
example, without either himself or (apparently) Socrates being 
completely aware of it. 

Still as late as 1800, influential philosophers such as Immanuel 
Kant held that Euclid's axio ms represented absolute truth. 

But do they? Would anyone question the statement that "the 
whole is greater than the pari"? Since 10 can be broken up into 
6 + 4, are we not completely right in assuming that 10 is greater 
than either 6 or 4? If an astronaut can get into a space capsule, 
would we not be right in assuming that the volume of the capsule 
is greater than the volume of the astronaut? How could we doubt 
the general truth of the axiom? 

Well, any list of consecutive numbers can be divided into odd 
numbers and even numbers, so that we might conclude that in 
any such list of consecutive numbers, the total of all numbers pres- 
ent must be greater than the total of even numbers. And yet if 
we consider an infinite list of consecutive numbers, it turns out 
that the total number of all the numbers is equal to the total num- 



ber of all the even numbers. In what is called "transfinite mathe- 
matics" the particular axiom about the whole being greater than 
the part simply does not apply. 

Again, suppose that two automobiles travel between points A 
and B by identical routes. The two routes coincide. Are they 
equal? Not necessarily. The first automobile traveled from A to B, 
while the second traveled from B to A. In other words, two lines 
might coincide and yet be unequal since the direction of one might 
be different from the direction of the other. 

Is this just fancy talk? Can a line be said to have direction? Yes, 
indeed. A line with direction is a "vector" and in "vector mathe- 
matics" the rules aren't quite the same as in ordinary mathematics 
and things can coincide without being equal. 

In short, then, axioms are not examples of absolute truth and 
it is very likely that there is no such thing as absolute truth at all. 
The axioms of Euclid are axioms not because they appear as ab- 
solute truth out of some inner enlightenment but only because 
they seem to be true in the context of the real world. 

And that is why the geometric theorems derived from Euclid's 
axioms seem to correspond with what we call reality. They started 
with what we call reality. 

It is possible to start with any set of axioms, provided they are 
not self-contradictory, and work up a system of theorems consist- 
ent with those axioms and with each other, even though they are 
not consistent with what we think of as the real world. This does 
not make the "arbitrary mathematics" less "true" than the one 
starting from Euclid's axioms, only less useful, perhaps. Indeed, 
an "arbitrary mathematics" may be more useful than ordinary 
"common-sense" mathematics in special regions such as those of 
transfinites or of vectors. 

Even so, we must not confuse "useful" and "true." Even if an 
axiomatic system is so bizarre as to be useful in no conceivable 
practical sense, we can nevertheless say nothing about its "truth." 
If it is self-consistent, that is all we have a right to demand of any 
system of thought. "Truth" and "reality" are theological words, 
not scientific ones. 


But back to Euclid's axioms. So far I have only listed the five 
"common notions." There were also five more axioms on the list 
that were specifically applicable to geometry and these were later 
called "postulates." The first of these postulates was: 

1) It is possible to draw a straight line from any point to any 
other point. 

This seems eminently acceptable, but are you sure? Can you 
prove that you can draw a line from the Earth to the Sun? If you 
could somehow stand on the Sun safely and hold the Earth mo- 
tionless in its orbit, and somehow stretch a string from the Earth 
to the Sun and pull it absolutely taut, that string would represent 
a straight line from Earth to Sun. You're sure that this is a reason- 
able "thought experiment" and I'm sure it is, too, but we only 
assume that matters can be so. We can't ever demonstrate them, 
or prove them mathematically. 

And, incidentally, what is a straight line? I have just made the 
assumption that if a string is pulled absolutely taut, it has a shape 
we would recognize as what we call a straight line. But what is 
that shape? We simply can't do better than say, "A straight line is 
something very, very thin and very, very straight," or, to paraphrase 
Gertrude Stein, "A straight line is a straight line is a straight 

Euclid defines a straight line as "a line which lies evenly with 
the points on itself," but I would hate to have to try to describe 
what he means by that statement to a student beginning the study 
of geometry. 

Another definition says that: A straight line is the shortest dis- 
tance between two points. 

But if a string is pulled absolutely taut, it cannot go from the 
point at one end to the point at the other in any shorter distance, 
so that to say that a straight line is the shortest distance between 
two points is the same as saying that it has the shape of an ab- 
solutely taut string, and we can still say "And what shape is that?" 

In modern geometry, straight lines are not defined at all. What 
is said, in essence, is this: Let us call something a line which has 
the following properties in connection with other undefined 



terms like "point," "plane," "between," "continuous," and so on. 
Then the properties are listed. 

Be that as it may, here are the remaining postulates of Euclid: 

2) A finite straight line can be extended continuously in a 
straight line. 

3) A circle can be described with any point as center and any 
distance as radius. 

4) All right angles are equal. 

5) If a straight line falling on two straight lines makes the in- 
terior angles on the same side less than two right angles, the two 
straight lines, if produced indefinitely, meet on that side on which 
are the angles less than the two right angles. 

I trust you notice something at once. Of all the ten axioms of 
Euclid, only one — the fifth postulate — is a long jawbreaker of a 
sentence; and only one — the fifth postulate — doesn't make instant 

Take any intelligent person who has studied arithmetic and 
who has heard of straight lines and circles and give him the ten 
axioms one by one and let him think a moment and he will say, 
"Of course!" to each of the first nine. Then recite the fifth postu- 
late and he will surely say, "What!" 

And it will take a long time before he understands what's going 
on. In fact, I wouldn't undertake to explain it myself without a 
diagram like the one on page 124. 

Consider two of the solid lines in the diagram: the one that 
runs from point C to point D through point M (call it line CD 
after the end points) and the one that runs through points G, 
L, and H (line GH). A third line, which runs through points A, 
L, M, and B (line AB), crosses both GH and CD, making an- 
gles with both. 

If line CD is supposed to be perfectly horizontal, and line AB 
is supposed to be perfectly vertical, then the four angles made in 
the crossing of the two lines (angles CMB, BMD, DML, and 
LMC) are right angles and are all equal (by postulate 4). In par- 




G A J < A 

E A " A V . I > AA F 


C M D 

ticular, angles DML and LMC, which 1 have numbered in the 
diagram as 3 and 4, are equal, and are both right angles. 

(I haven't bothered to define "perfectly horizontal" or "per- 
fectly vertical" or "crosses" or to explain why the crossing of a per- 
fectly horizontal line with a perfectly vertical line produces four 
right angles, but 1 am making no pretense of being completely 
rigorous. This sort of thing can be made rigorous but only at the 
expense of a lot more talk than I am prepared to give.) 

Now consider line GH. It is not perfectly horizontal. That 
means the angles it produces at its intersection (I haven't de- 
fined "intersection") with line AB are not right angles and are not 
all equal. It can be shown that angles ALH and GLB are equal 
and that angles HLB and GLA are equal but that either of the 
first pair is not equal to either of the second pair. In particular, 
angle GLB (labeled 2) is not equal to angle HLB (labeled 1). 

Suppose we draw line EF, passing through L, and that line EF 
is (like line CD) perfectly horizontal. In that case it makes four 
equal right angles at its intersection with line AB. In particular, 
angles FLB and ELB are right angles. But angle HLB is con- 
tained within angle FLB (what does "is contained within" mean?) 
with room to spare. Since angle HLB is only part of FLB and the 



latter is a right angle then angle HLB (angle 1) is less than a 
right angle, by the fifth "common notion." 

In the same way, by comparing angle ELB, known to be a right 
angle, with angle GLB (angle 2), we can show that angle 2 is 
greater than a right angle. 

The "interior angles" of the diagram are those on the side of 
line GH that faces line CD, and those on the side of line CD that 
faces line GH. In other words, they are angles 1, 2, 3, and 4. 

The fifth postulate talks about "the interior angles on the same 
side," that is, 1 and 4 on one side and 2 and 3 on the other. Since 
we know that 3 and 4 are right angles, that 1 is less than a right 
angle, and that 2 is more than a right angle, we can say that the 
interior angles on one side, 1 and 4, have a sum less than two 
right angles, while the interior angles on the other have a sum 
greater than two right angles. 

The fifth postulate now states that if the lines GH and CD are 
extended, they will intersect on the side where the interior angles 
with a sum less than two right angles are located. And, indeed, if 
you look at the diagram you will see that if lines GH and CD are 
extended on both sides (dotted lines), they will intersect at point 
N on the side of interior angles 1 and 4. On the other side, they 
just move farther and farther apart and clearly will never intersect. 

On the other hand, if you draw line JK through L, you would 
reverse the situation. Angle 2 would be less than a right angle and 
angle 1 would be greater than a right angle (where angle 2 is now 
angle JLB and angle 1 is now angle KLB). In that case interior 
angles 2 and 3 would have a sum less than two right angles and 
interior angles 1 and 4 would have a sum greater than two right 
angles. If lines JK and CD were extended (dotted lines), they 
would intersect at point O on the side of interior angles 2 and 
3. On the other side they would merely diverge further and further. 

Now that I've explained the fifth postulate at great length (and 
even then only at the cost of being very un-rigorous) you might 
be willing to say, "Oh yes, of course. Certainly! It's obvious!" 

Maybe, but if something is obvious, it shouldn't require hun- 


dreds of words of explanation. I didn't have to belabor any of the 
other nine axioms, did I? 

Then again, having explained the fifth postulate, have I proved 
it? No, I have only interpreted the meaning of the words and 
then pointed to the diagram and said, "And indeed, if you look 
at the diagram, you will see — " 

But that's only one diagram. And it deals with a perfectly verti- 
cal line crossing two lines of which one is perfectly horizontal. 
And what if none of the lines are either vertical or horizontal 
and none of the interior angles are right angles? The fifth postu- 
late applies to any line crossing any two lines and I certainly 
haven't proved that. 

I can draw a million diagrams of different types and show that 
in each specific case the postulate holds, but that is not enough. 
I must show that it holds in every conceivable case, and this can't 
be done by diagrams. A diagram can only make the proof clear; 
the proof itself must be derived by permissible logic from more 
basic premises already proved, or assumed. This I have not done. 

Now let's consider the fifth postulate from the standpoint of 
moving lines. Suppose line GH is swiveled about L as a pivot in 
such a way that it comes closer and closer to coinciding with line 
EF. (Does a straight line remain a straight line while it swivels 
in this fashion? We can only assume it does.) As line GH swivels 
toward line EF, the point of intersection with line CD (point N) 
moves farther and farther to the right. 

If you started with line JK and swiveled it so that it would even- 
tually coincide with line EF, the intersection point O would move 
off farther and farther to the left. If you consider the diagram 
and make a few markings on it (if you have to) you will see this 
for yourself. 

But consider line EF itself. When GH has finally swiveled so as 
to coincide with line EF, we might say that intersection point N 
has moved off an infinite distance to the right (whatever we mean 
by "infinite distance") and when line JK coincides with line EF, 
the intersection point O has moved off an infinite distance to the 



left. Therefore, we can say that line EF and line CD intersect at 
two points, one an infinite distance to the right and one an in- 
finite distance to the left. 

Or let us look at it another way. Line EF, being perfectly hori- 
zontal, intersects line AB to make four equal right angles. In that 
case, angles 1, 2, 3, and 4 are all right angles and all equal. Angles 

1 and 4 have a sum equal to two right angles, and so do angles 

2 and 3. 

But the fifth postulate says the intersection comes on the side 
where the two interior angles have a sum less than two right an- 
gles. In the case of lines EF and CD crossed by line AB, neither 
set of interior angles has a sum less than two right angles and 
there can be an intersection on neither side. 

We have now, by two sets of arguments, demonstrated first that 
lines EF and CD intersect at two points, each located an infinite 
distance away, and second that lines EF and CD do not intersect 
at all. Have we found a contradiction and thus shown that there 
is something wrong with Euclid's set of axioms? 

To avoid a contradiction, we can say that having an intersec- 
tion at an infinite distance is equivalent to saying there is no in- 
tersection. They are different ways of saying the same thing. To 
agree that "saying a" is equal to "saying b" in this case is consist- 
ent with all the rest of geometry, so we can get away with it. 

Let us now say that two lines, such as EF and CD, which do not 
intersect with each other when extended any finite distance, how- 
ever great, are "parallel." 

Clearly, there is only one line passing through L that can be 
parallel to line CD, and that is line EF. Any line through L that 
does not coincide with line EF is (however slightly) either of the 
type of line GH or of line JK, with an interior angle on one side 
or the other that is less than a right angle. This argument is sleight 
of hand, and not rigorous, but it allows us to see the point and 
say: Given a straight line, and a point outside that line, it is pos- 
sible to draw one and only one straight line through that point 
parallel to the given line. 

This statement is entirely equivalent to Euclid's fifth postulate. 


If Euclid's fifth postulate is removed and this statement put in 
its place, the entire structure of Euclidean geometry remains 
standing without as much as a quiver. 

The version of the postulate that refers to parallel lines sounds 
clearer and easier to understand than the way Euclid puts it, be- 
cause even the beginning student has some notion of what paral- 
lel lines look like, whereas he may not have the foggiest idea of 
what interior angles are. That is why it is in this "parallel" form 
that you usually see the postulate in elementary geometry books. 

Actually, though, it isn't really simpler and clearer in this form, 
for as soon as you try to explain what you mean by "parallel" 
you're going to run into the matter of interior angles. Or, if you 
try to avoid that, you'll run into the problem of talking about 
lines of infinite length, of intersections at an infinite distance be- 
ing equivalent to no intersection, and that's even worse. 

But look, just because I didn't prove the fifth postulate doesn't 
mean it can't be proven. Perhaps by some line of argument, ex- 
ceedingly lengthy, subtle and ingenious, it is possible to prove 
the fifth postulate by use of the other four postulates and the five 
common notions (or by use of some additional axiom not in- 
cluded in the list which, however, is much simpler and more 
"obvious" than the fifth postulate is). 

Alas, no. For two thousand years mathematicians have now and 
then tried to prove the fifth postulate from the other axioms 
simply because that cursed fifth postulate was so long and so un- 
obvious that it didn't seem possible that it could be an axiom. 
Well, they always failed and it seems certain they must fail. The 
fifth postulate is just not contained in the other axioms or in any 
list of axioms useful in geometry and simpler than itself. 

It can be argued, in fact, that the fifth postulate is Euclid's 
greatest achievement. By some remarkable leap of insight, he 
realized that, given the nine brief and clearly "obvious" axioms, 
he could not prove the fifth postulate and that he could not do 
without it either, and that, therefore, long and complicated though 
the fifth postulate was, he had to include it among his assumptions. 



So for two thousand years the fifth postulate stood there: long, 
ungainly, puzzling. It was like a flaw in perfection, a standing 
reproach to a line of argument otherwise infinitely stately. It 
bothered the very devil out of mathematicians. 

And then, in 1733, an Italian priest, Girolamo Saccheri, got 
the most brilliant notion concerning the fifth postulate that any- 
one had had since the time of Euclid, but wasn't brilliant enough 
himself to handle it- 

Let's go into that in the next chapter. 


1 1 

There are occasionally problems in immersing myself in these 
science essays I write. For instance, I watched a luncheon com- 
panion sprinkle salt on his dish after an unsatisfactory forkful, 
try another bite, and say with satisfaction, "That's much better." 

I stirred uneasily and said, "Actually, what you mean is, 'I like 
that much better.' In saying merely, "That's much better,' you are 
making the unwarranted assumption that food can be objectively 
better or worse in taste and the further assumption that your own 
subjective sensation of taste is a sure guide to the objective 

I think I came within a quarter of an inch of getting that dish, 
salted to perfection as it was, right in the face; and would have 
well deserved it, too. The trouble, you see, was that I had just 
written the previous chapter and was brimful on the subject of 

So let's get back to that. The subject under consideration is 
Euclid's "fifth postulate," which I will repeat here so that you 
won't have to refer back to it: 

If a straight line falling on two straight lines makes the interior 
angles on the same side less than two right angles, the two straight 
lines, if produced indefinitely, meet on that side on which are the 
angles less than the two right angles. 

All Euclid's other axioms are extremely simple but he appar- 
ently realized that this fifth postulate, complicated as it seemed, 
could not be proved from the other axioms, and must therefore 
be included as an axiom itself. 

For two thousand years after Euclid other geometers kept try- 



ing to prove Euclid too hasty in having given up, and shove to 
find some ingenious method of proving the fifth postulate horn 
the other axioms, so that it might therefore be removed from the 
list — if only because it was too long, too complicated, and too not 
immediately obvious to seem a good axiom. 

One system of approaching the problem was to consider the 
following quadrilateral: 

Two of the angles, DAB and ABC are given as right angles in 
this quadrilateral, and side AD is equal in length to side BC. 
Given these facts, it is possible to prove that side DC is equal to 
side AB and that angles ADC and DCB are also right angles (so 
that the quadrilateral is actually a rectangle) if Euclid's fifth 
postulate is used. 

If Euclid's fifth postulate is not used, then by using only the 
other axioms, all one can do is to prove that angles ADC and 
DCB are equal, but not that they are actually right angles. 

The problem then arises whether it is possible to show that 
from the fact that angles ADC and DCB are equal, it is possible 
to show that they are also right angles. If one could do that, it 
would then follow from the fact that quadrilateral ABCD is a 
rectangle, that the fifth postulate is true. This would have been 
proven from the other axioms only and it would no longer be 
necessary to include Euclid's fifth among them. 

Such an attempt was first made by the medieval Arabs, who 
carried on the traditions of Greek geometiy while Western Eu- 
rope was sunk in darkness. The first to draw this quadrilateral and 


labor over its right angles was none other than Omar Khayyam 


Omar pointed out that if angles ADC and DCB were equal, 
then there were three possibilities: 1) they were each a right 
angle, 2) they were each less than a right angle, that is "acute," 
or 3) they were each more than a right angle, or "obtuse." 

He then went through a line of argument to show that the 
acute and obtuse cases were absurd, based on the assumption 
that two converging lines must intersect. 

To be sure, it is perfectly conrmonsensical to suppose that two 
converging lines must intersect, but, unfortunately, conmronsense 
or not, that assumption is mathematically equivalent to Euclid's 
fifth postulate. Omar Khayyam ended, therefore, by "proving" the 
fifth postulate by assuming it to be true as one of the conditions 
of the proof. This is called either "arguing in a circle" or "begging 
the question," but whatever it is called, it is not allowed in 

Another Arabian mathematician, Nasir Eddin al-Tus (1201- 
74), made a similar attempt on the quadrilateral, using a differ- 
ent and more complicated assumption to outlaw the acute and 
obtuse cases. Alas, his assumption was also mathematically equiva- 
lent to Euclid's fifth. 

Which brings us down to the Italian, Girolamo Saccheri (1667- 
1733), whom I referred to at the end of the previous chapter and 
who was both a professor of mathematics at the University of Pisa, 
and a Jesuit priest. 

He knew of Nash Eddin's work and he, too, tackled the quadri- 
lateral. Saccheri, however, introduced something altogether new, 
something that in two thousand years no one had thought of do- 
ing in connection with Euclid's fifth. 

Until then, people had omitted Euclid's fifth to see what would 
happen, or else had made assumptions that turned out to be 

* He wrote clever quatrains which Edward FitzGerald even more cleverly 
translated into English in 1859, making Omar forever famous as a hedo- 
nistic and agnostic poet, but the fact is that he ought to be remembered as 
a great mathematician and astronomer. 


equivalent to Euclid's fifth. What Saccheri did was to begin by 
assuming Euclid's fifth to be false, and to substitute for it some 
other postulate that was contradictory to it. He planned then to 
by to build up a geometry based on Euclid's other axioms plus 
the "alternate fifth" until he came to a contradiction (proving 
that a particular theorem was both true and false, for instance). 

When the contradiction was reached, the "alternate fifth" 
would have to be thrown out. If every possible "alternate fifth" is 
eliminated in this fashion, then Euclid's fifth must be true. This 
method of proving a theorem by showing all other possibilities to 
be absurd is a perfectly acceptable mathematical technique* and 
Saccheri was on the right road. 

Working on this system, Saccheri therefore stalled by assuming 
that the angles ADC and DCB were both greater than a right 
angle. With this assumption, plus all the axioms of Euclid other 
than the fifth, he began working his way through what we might 
call "obtuse geometry." Quickly, he came across a contradiction. 
This meant that obtuse geometry could not be true and that 
angles ADC and DCB could not each be greater than a right 

This accomplishment was so important that the quadrilateral 
which Omar Khayyam had first used in connection with Euclid's 
fifth is now called the "Saccheri quadrilateral." 

Greatly cheered by this, Saccheri then tackled "acute geome- 
try," beginning with the assumption that angles ADC and DCB 
were each smaller than a right angle. He must have begun the 
task lightheartedly, sure that, as in the case of obtuse geometry, 
he would quickly find a contradiction in acute geometry. If that 
were so, Euclid's fifth would stand proven and his "right-angle 
geometry" would no longer require that uncomfortably long state- 
ment as an axiom. 

As Saccheri went on from proposition to proposition in his 
acute geometry, his feeling of pleasure gave way to increasing 

* This is equivalent to Sherlock Holmes's famous dictum that when the 
impossible has been eliminated, whatever remains, however improbable, must 
be true. 




anxiety, for he did not come across any contradiction. More and 
more he found himself faced with the possibility that one could 
build up a thoroughly self-consistent geometry which was based 
on at least one axiom that directly contradicted a Euclidean 
axiom. The result would be a "non-Euclidean” geometry which 
might seem against common sense, but which would be internally 
self-consistent and therefore mathematically valid. 

For a moment, Saccheri hovered on the very brink of mathe- 
matical immortality and — backed away. 

He couldn't! To accept the notion of a non-Euclidean geome- 
try took too much courage. So mistakenly had scholars come to 
confuse Euclidean geometry with absolute truth, that any refuta- 
tion of Euclid would have roused the deepest stirrings of anxiety 
in the hearts and minds of Europe's intellectuals. To doubt 
Euclid was to doubt absolute truth and if there was no absolute 
truth in Euclid, might it not be quickly deduced that there was 
no absolute truth anywhere? And since the firmest claim to abso- 
lute truth came from religion, might not an attack on Euclid be 
interpreted as an attack on God? 

Saccheri was clearly a mathematician of great potential, but he 
was also a Jesuit priest and a human being, so his courage failed 
him and he made the great denial.* When his gradual develop- 
ment of acute geometry went on to the point where he could take 
it no longer, he argued himself into imagining he had found an 
inconsistency where, in fact, he hadn't, and with great relief, he 
concluded that he had proved Euclid's fifth. In 1733, he published 
a book on his findings entitled (in English) Euclid Cleared of 
Every Flaw, and, in that same year, died. 

By his denial Saccheri had lost immortality and chosen oblivion. 
His book went virtually unnoticed until attention was called to 
it by a later Italian mathematician, Eugenio Beltrami (1835 — 
1900), after Saccheri's failure had been made good by others. Now 
what we know of Saccheri is just this: that he had his finger on a 

* I am not blaming him. Placed in his position, I would undoubtedly 
have done the same. It's just too bad, that's all. 


major mathematical discovery a century before anybody else and 
had lacked the guts to keep his finger firmly on it. 

Let us next move forward nearly a century to the German 
mathematician Karl F. Gauss (1777-1855). It can easily be argued 
that Gauss was the greatest mathematician who ever lived. Even 
as a young man he astonished Europe and the scientific world 
with his brilliance. 

He considered Euclid's fifth about 1815 and came to the same 
conclusion to which Euclid had come — that the fifth had to be 
made an axiom because it couldn't be proved from the other ax- 
ioms. Gauss further came to the conclusion from which Saccheri 
had shrunk away — that there were other self-consistent geometries 
which were non-Euclidean, in that an alternate axiom replaced 
the fifth. 

And then he lacked the guts to publish, too, and here I disclaim 
sympathy. The situation was different. Gauss had infinitely more 
reputation than Saccheri; Gauss was not a priest; Gauss lived in 
a land where, and at a time when, the hold of the Church was less 
to be feared. Gauss, genius or not, was just a coward. 

Which brings us to the Russian mathematician Nikolai Ivano- 
vich Lobachevski (1793-1856).* In 1826, Lobachevski also began 
to wonder if a geometry might not be non-Euclidean and yet con- 
sistent. With that in mind, he worked out the theorems of "acute 
geometry" as Saccheri had done a century earlier, but in 1829, 
Lobachevski did what neither Saccheri nor Gauss had done. He 
did not back away and he did publish. Unfortunately, what he 
published was an article in Russian called "On the Principles of 
Geometry" in a local periodical (he worked at the University of 
Kazan, deep in provincial Russia). 

Who reads Russian? Lobachevski remained largely unknown. 
It wasn't until 1840 that he published his work in German and 

* Nikolai Ivanovich Lobachevski is mentioned in one of Tom Lehrer's 
satiric songs and to any Tom Lehrer fan (like myself) it seems strange to 
see the name mentioned in a serious connection, but Lehrer is a mathema- 
tician by trade and he made use of a real name. 


brought himself to the attention of the world of mathematics 

Meanwhile, though, a Hungarian mathematician, Janos Bolyai 
(1802-60), was doing much the same thing. Bolyai is one of the 
most romantic figures in the history of mathematics since he also 
specialized in such things as the violin and the dueling sword — in 
the true tradition of the Hungarian aristocrat. There is a story 
that he once fenced with thirteen swordsmen one after the other, 
vanquishing them all — and playing the violin between bouts. 

In 1831, Bolyai's father published a book on mathematics. 
Young Bolyai had been pondering over Euclid's fifth for a number 
of years and now he persuaded his father to include a twenty-six- 
page appendix in which the principles of acute geometry were 
described. It was two years after Lobachevski had published but 
as yet no one had heard of the Russian and nowadays, Lobachev- 
ski and Bolyai generally share the credit for having discovered 
non-Euclidean geometry. 

Since the Bolyais published in German, Gauss was at once 
aware of the material. His commendation would have meant a 
great deal to the young Bolyai. Gauss still lacked the courage to 
put his approval into print, but he did praise Bolyai's work ver- 
bally. And then, he couldn't resist — He told Bolyai he had had the 
same ideas years before but hadn't published, and showed him 
the work. 

Gauss didn't have to do that. His reputation was unshakable; 
even without non-Euclidean geometry, he had done enough for a 
dozen mathematicians. Since he had lacked the courage to pub- 
lish, he might have had the decency to let Bolyai take the credit. 
But he didn't. Genius or not, Gauss was a mean man in some 

Poor Bolyai was so embarrassed and humiliated by Gauss's dis- 
closure, that he never did any further work in mathematics. 

And what about obtuse geometry? Saccheri had investigated 
that and found himself enmeshed in contradiction, so that had 
been thrown out. Still, once the validity of non-Euclidean geome- 
try had been established, was there no way of rehabilitating obtuse 
geometry, too? 


Yes, there was — but only at the cost of making a still more radi- 
cal break with Euclid. Saccheri, in investigating obtuse geometry 
had made use of an unspoken assumption that Euclid himself had 
also used — that a line could be infinite in length. This assump- 
tion introduced no contradiction in acute geometry or in right- 
angle geometry (Euclid's), but it did create trouble in obtuse 

But then, drop that too. Suppose that, regardless of "common 
sense" you were to make the assumption that any line had to 
have some maximum finite length. In that case all the contradic- 
tion in obtuse geometry disappeared and there was a second valid 
variety of non-Euclidean geometry. This was first shown in 1854 
by the German mathematician Georg F. Riemann (1826-66). 

So now we have three kinds of geometry, which we can dis- 
tinguish by using statements that are equivalent to the variety of 
fifth postulate used in each case: 

A) Acute geometry (non-Euclidean): Through a point not on 
a given line, an infinite number of lines parallel to the given line 
may be drawn. 

B) Right-angle geometry (Euclidean): Through a point not 
on a given line, one and only one line parallel to the given line 
may be drawn. 

C) Obtuse geometry (non-Euclidean): Through a point not 
on a given line, no lines parallel to the given line may be drawn. 

You can make the distinction in another and equivalent way: 

A) Acute geometry (non-Euclidean): The angles of a triangle 
have a sum less than 180°. 

B) Right-angle geometry (Euclidean): The angles of a triangle 
have a sum exactly equal to 180°. 

C) Obtuse geometry (non-Euclidean): The angles of a tri- 
angle have a sum greater than 180°. 

You may now ask: But which geometry is true? 

If we define "true" as internally self-consistent, then all three 
geometries are equally true. 

Of course, they are inconsistent with each other and perhaps 


only one corresponds with reality. We might therefore ask: Which 
geometry corresponds to the properties of the real universe? 

The answer is, again, that all do. 

Let us, for instance, consider the problem of traveling from 
point A on Earth's surface to point B on Earth's surface, and sup- 
pose we want to go from A to B in such a way as to traverse the 
least distance. 

In order to simplify the results, let us make two assumptions. 
First, let us assume that the Earth is a perfectly smooth sphere. 
This is almost true, as a matter of fact, and we can eliminate 
mountains and valleys and even the equatorial bulge without too 
much distortion. 

Second, let us assume that we are confined in our travels to 
the surface of the sphere and cannot, for instance, burrow into 
its depth. 

In order to determine the shortest distance from A to B on 
the surface of the Earth, we might stretch a thread from one point 
to the other and pull it taut. If we were to do this between two 
points on a plane, that is, on a surface like that of a flat blackboard 
extended infinitely in all directions, the result would be what we 
ordinarily call a "straight line." 

On the surface of the sphere, the result, however, is a curve, 
and yet that curve is the analogue of a straight line, since that 
curve is the shortest distance between two points on the surface of 
a sphere. There is difficulty in forcing ourselves to accept a curve 
as analogous to a straight line because we've been thinking 
"straight" all our lives. Let us use a different word, then. Let us 
call the shortest distance between two points on any given surface 
a "geodesic."* 

On a plane, a geodesic is a straight line; on a sphere, a geodesic 
is a curve, and, specifically, the arc of a "great circle." Such a great 
circle has a length equal to the circumference of the sphere and 
lies in a plane that passes through the center of the sphere. On the 

* "Geodesic" is from Greek words meaning "to divide the Earth" because 
any geodesic on the face of the Earth, if extended as far as possible, divides 
the surface of the Earth into two equal parts. 


Earth, the equator is an example of a great circle and so are all the 
meridians. There are an infinite number of great circles that can 
be drawn on the surface of any sphere. If you choose any pair of 
points on a sphere and connect each pair by a thread which is 
pulled taut, you have in each case the arc of a different great circle. 

You can see that on the surface of a sphere, there is no such 
thing as a geodesic of infinite length. If it is extended, it simply 
meets itself as it goes around the sphere and becomes a closed 
curve. On the surface of the Earth, a geodesic can be no longer 
than 25,000 miles. 

Furthermore, any two geodesies drawn on a sphere intersect 
if produced indefinitely, and do so at two points. On the surface 
of the Earth, for instance, any two meridians meet at the north 
pole and the south pole. This means that, on the surface of 
a sphere, through any point not on a given geodesic, no geodesic 
can be drawn parallel to the given geodesic. No geodesic can be 
drawn through the point that won't sooner or later intersect the 
given geodesic. 

Then, too, if you draw a triangle on the surface of a sphere, 
with each side the arc of a great circle, then the angles will have 
a sum greater than 180°. If you own a globe, imagine a triangle 
with one of its vertices at the north pole, with a second at the 
equator and io° west longitude, and the third at the equator and 
100° west longitude. You will find that you will have an equi- 
lateral triangle with each one of its angles equal to 90°. The sum 
of the three angles is 270 °. 

This is precisely the geometry that Riemann worked out, if the 
geodesies are considered the analogues of straight lines. It is a 
geometry of finite lines, no parallels, and triangular angle-sums 
greater than 180°. What we have been calling "obtuse geometry" 
then might also be called "sphere geometry." And what we have 
been calling "right-angle geometry" or "Euclidean geometry" 
might also be called "plane geometry." 

In 1865, Eugenio Beltrami drew attention to a shape called a 
'pseudosphere," which looks like two cornets joined wide mouth 


to wide mouth, and with each comet extending infinitely out in 
either direction, ever narrowing but never quite closing. The geo- 
desies drawn on the surface of a pseudosphere fulfill the require- 
ments of acute geometry. 

Geodesies on a pseudosphere are infinitely long and it is pos- 
sible for two particular geodesies to be extended indefinitely with- 
out intersecting and therefore to be parallel. In fact, it is possible 
to draw two geodesies on the surface of a pseudosphere that do 
intersect and yet have neither one intersecting a third geodesic 
lying outside the two.* In fact, since an infinite number of geo- 
desies can be drawn in between the two intersecting geodesies, 
all intersecting in the same point, there are an infinite number 
of possible geodesies through a point, all of which are parallel to 
another geodesic not passing through the point. 

In other words "acute geometry" can be looked at as "pseudo- 
sphere geometry." 

But now — granted that all three geometries are equally valid 
under circumstances suiting each — which is the best description 
of the universe as a whole? 

This is not always easy to tell. If you draw a triangle with geo- 
desies of a given length on a small sphere and then again on a 
large sphere, the sum of the angles of the triangle will be greater 
than 180° in either case, but the amount by which it is greater 
will be greater in the case of the small sphere. 

If you imagine a sphere growing larger and larger, a triangle of 
a given size on its surface will have an angle-sum closer and closer 
to 180° and eventually even the most refined possible measure- 
ment won't detect the difference. In short, a small section of a 
very large sphere is almost as flat as a plane and it becomes im- 
possible to tell the difference. 

This is true of the Barth, for instance. It is because the Barth 
is so large a sphere that small parts of it look flat and that it took 

* This sounds nonsensical because we are used to thinking in terms of 
planes where the geodesies are straight lines and where two intersecting lines 
cannot possibly be both parallel to a third line. On a pseudosphere, the 
geodesies curve, and curve in such a way as to make the two parallels pos- 



so long for mankind to satisfy himself that it was spherical despite 
the fact that it looked flat. 

Well, there is a similar problem in connection with the universe 

Light travels from point to point in space; from the Sun to the 
Earth, or from one distant Galaxy to another, over distances 
many times those possible on Earth's surface. 

We assume that light in traveling across the parsecs travels in 
a straight line but, of course, it really travels in a geodesic, which 
may or may not be a straight line. If the universe obeys Euclidean 
geometry, the geodesic is a straight line. If the universe obeys 
some non-Euclidean geometry, then the geodesies are curves of 
one sort or another. 

It occurred to Gauss to form triangles with beams of light 
traveling through space from one mountaintop to another, and 
measure the sum of the angles so obtained. To be sure, the sums 
turned out to be just about 180°, but were they exactly 180°? 
That was impossible to tell. If the universe were a sphere millions 
of light-years in diameter and if the light beams followed the curv- 
ings of such a sphere, no conceivable direct measurement possible 
today could detect the tiny amount by which the angle sum ex- 
ceeded 180°. 

In 1916, however, Einstein worked out the General Theory of 
Relativity, and found that in order to explain the workings of 
gravitation, he had to assume a universe in which light (and every- 
thing else) traveled in non-Euclidean geodesies. 

By Einstein's theory, the universe is non-Euclidean and is, in 
fact, an example of "obtuse geometry." 

To put it briefly, then. Euclidean geometry, far from being the 
absolute and eternal verity it was assumed to be for two thousand 
years, is only the highly restricted and abstract geometry of the 
plane, and one that is merely an approximation of the geometry 
of such important things as the universe and the Earth's surface. 

It is not the plain truth so many have taken for granted it was — 
but only the plane truth.* 

* Well, J think it's clever. 

D — The Problem of the Platypus 


A friend said to me once that he would love to see my filing sys- 
tem. So I took him to my office and said, "This file is for corre- 
spondence. Here I keep old manuscripts. Here I have manuscripts 
in preparation. This is the card file of my books — of my shorter 
fiction — of my shorter non-fiction — " 

"No, no, no," he said. "That's all trivial. Where do you keep 
your reference files?" 

"What reference files?" said I, blankly. (I very often say things 
blankly. I think it's part of my charm — or maybe naivete.) 

"The cards on which you list items you may need for future 
articles or books and then file them according to various subjects." 

"I don't do that," I said, growing anxious. "Am I supposed to?" 

"But how do you keep things straight in your head, then?" 

I was glad to be able to answer that one definitely. "I don't 
know," I said, and he seemed pretty annoyed with me. 

Well, I don't. All 1 know is that I've been a classifier ever since 
I can remember. Everything falls into categories with me. Every- 
thing is divided and counted up and put into neat stacks in my 
mind. I don't worry about it; it happens by itself. 

Of course, I sometimes worry about the details. For instance, 
what with one thing or another, the actual number of books I 
have published has become an issue. I am forever being asked, 
"How many books have you published?"* 

But what's a book? 

Recently, the second edition of my book The Universe was 
published. Do I count that as a new book? Of course not. It's 
updated but the updating doesn't represent enough in the way of 

* The answer is 117 at the moment of writing, if you are dying of 


change to make me consider the book "new." On the other hand, 
later, the third edition of my book The Intelligent Man's Guide 
To Science is being published. I counted the second edition as a 
new book and I intend to count the third edition as such, because 
in each case the changes introduced were substantial and as time- 
and energy-consuming as a new book would have been. 

You might think all this is something I can chop and change to 
please myself, but not exactly. In my book Opus 100, I listed my 
first hundred books in chronological order and that list became 
"official." But is it correct? Was I right in omitting this or that 
item from the list or, for that matter, including this item or that. 

Unimportant? Sure, but it does help me sympathize with those 
classifiers who involve themselves with more intricate matters than 
a listing of books. For instance- 

How do you tell a mammal from a reptile? 

The easiest and quickest way is to decide that a mammal is 
covered with hair and a reptile is covered with scales. Of course, 
you have to be liberal in making this distinction. Some organisms 
we consider mammals don't have very much hair. Human beings 
don't — but we have some hair. Elephants have even less, but they 
have some. Whales have still less, but even they have some. Dol- 
phins usually have anywhere from two to eight hairs near the tip 
of the snout. Even in those whales where hair is altogether absent, 
it is present at some time in the fetal development. 

And one hair is, in this respect, as good as a million, for any 
hair at all is the hallmark of the mammal. No creature that we 
consider to be definitely a non-mammal has even one true hair. 
They may have structures that look like hair, but the resemblance 
disappears if we consider its microscopic structure, its chemical 
makeup, its anatomical origin, or all three. 

A somewhat less useful distinction is that mammals (well, 
most) bring forth their young alive, while reptiles (well, most) 
don't. Some reptiles, such as sea snakes, bring forth living young, 
but in doing so they merely retain the eggs within their body till 
they hatch. The developing embryos find their food within the 



egg and the fact that the egg is within the body is a point for se- 
curity, but not for nourishment. 

Mammals, on the other hand, or most of them, feed the de- 
veloping young out of the maternal bloodstream by means of an 
organ called the "placenta," in which the mother's blood vessels 
and the blood vessels of the embryo come close enough to allow 
molecules to seep across: food from mother to embryo, wastes 
from embryo to mother. (There is no actual joining of blood- 
streams, however.) 

A minority of mammals bring forth living, but very poorly de- 
veloped, young, and these must then continue their development 
in a special maternal pouch outside the body. A still smaller mi- 
nority of mammals lay eggs. But even the egg-layers have hair. 

Another point is that mammals feed their newborn young on 
milk secreted by special maternal glands. This is true even of the 
non-placental mammals; even of the egg-layers. And this is not 
true for any animal without hair (not one!). Milk seems to be a 
purely mammalian product and it is this, more than anything 
else, which seems to have impressed the classifiers. The very word 
"mammal" is from the Latin mamma, meaning "breast." 

Then, too, mammals maintain a constant internal temperature 
even though the environmental temperature may vary widely. 
Reptiles, on the other hand, have an internal temperature that 
tends, more or less, to match that of the environment. Since 
the internal temperature of mammals is close to ioo° F. and is 
therefore generally higher than the environmental temperature, 
mammals feel warm to the touch while reptiles feel, by compari- 
son, cold. That is why we speak of mammals as "warm-blooded" 
and reptiles as "cold-blooded," missing the essential point that 
the internal temperature is constant in the former case and in- 
constant in the latter. 

(To be sure, birds are warm-blooded, too, but there is no dan- 
ger in confusing a bird and a mammal. All birds, without excep- 
tion, have feathers and all non-birds, without exception, do not 
have feathers. — And except for birds and mammals, all organisms 
are cold-blooded.) 


I have by no means listed all the differences between mammals 
and reptiles, only those that the non-biologist can tell by looking 
at the creatures from a distance. If we want to indulge in dissec- 
tion, we can discover others. For instance, mammals have a flat 
muscle called the "diaphragm" which divides the chest from the 
abdomen. The diaphragm as it contracts, increases the volume of 
the chest cavity (at the expense of the abdominal cavity, which 
doesn't care) and helps draw air into the lungs. Reptiles do not 
have a diaphragm. In fact, no non-hairy organism does. 

So far, so good. But now we pass on to extinct creatures which 
biologists can study only in fossil form. Paleontologists (those 
biologists specializing in extinct species) have no hesitation in 
looking at a fossil and saying that it is reptilian or that it is mam- 
malian. The question at once arises: How? 

All the really obvious distinctions can't be used since, in gen- 
eral, all that the fossils offer us are the remains of what used to 
be bones and teeth. You can't look at a handful of bones and 
teeth and find traces of hair or breasts or milk or placentae or 

All you can do is compare the bones and teeth with those of 
modern reptiles and mammals and see if there are strictly hard- 
tissue distinctions. Then, you might assume that if an extinct 
creature had bones characteristic of mammals, it must also have 
had hair, breasts, a diaphragm, and the rest. 

Consider the skull. In the most primitive and earliest reptiles, 
the skull behind the eye was solid bone and on the other side of 
the bone were the jaw muscles. There was a tendency, however, to 
expose the jaw muscles and give them freer play, so that many 
reptiles had openings in the skull bounded by bony arches. The 
loss in sheer defensive strength was more than made up for by the 
improvement in the offense represented by larger, stronger jaws 
that could go snap! more firmly. On the balance, then, the rep- 
tiles which happened to develop these openings passed on to 
greater things. 

(Yet evolutionary "advances" are never universal, and never 


the only answer. One group of reptiles that had no use for a hole 
in the head, managed to survive for hundreds of millions of years 
and flourish, after a fashion, even today, though many, many hole- 
in-the-head groups have vanished. I'm talking about the turtles, 
whose jaw muscles are hidden under a solid wall of bone.) 

The reptiles developed openings in their skulls in a variety of 
patterns and, indeed, are classified into groups according to those 
patterns. This is not because this pattern is of overwhelming 
physiological importance in itself, but only because it is conven- 
ient, since if you have any part at all of a reptile, however long 
dead, you are likely to have its skull. 

But what about mammals, which are descended from reptiles? 
They have a single opening on either side of the skull just behind 
the eye bounded on the bottom by a narrow bony arch called the 
"zygomatic arch." 

So the paleontologist can look at a skull and from the nature 
of the openings tell at once whether it is reptilian or mammalian. 

Then, again, the lower jaw of a reptile is made up of seven dif- 
ferent bones, fused tightly into a strong structure. The lower 
jaw of a mammal is a single bone. (Some of the missing bones 
developed into the tiny bones of the middle ear. This is not as 
strange as it sounds. If you put your finger at the point where 
lower jaw meets upper jaw and where the old reptilian bones 
existed, you will find you are not very far from your ear.) 

As for the teeth, those of reptiles tended to be undifferentiated 
and all alike, of conelike structure. In mammals, the teeth are 
highly differentiated, cutting incisors in front, grinding molars in 
back, with tearing canines and premolars between. 

Since mammals evolved from reptilian forebears is there any 
way of recognizing which group of reptiles possessed the distinc- 
tion of being our ancestors? Certainly no living group of reptiles 
seems to be descended from anything mammalian or even ap- 
proaching the mammalian. We must look for some group that 
left no reptilian descendants at all. 

One such group, now entirely extinct (as reptiles), is called the 


"Synapsida." These had a single skull opening on either side of the 
head and included members who showed the clear beginnings of 

There were two important groups of synapsids. The earlier, dat- 
ing back some three hundred million years, were members of the 
order "Pelycosauria." The pelycosaurs are interesting chiefly be- 
cause their skulls seem to show the beginning of a zygomatic arch. 
Furthermore, their teeth show some differentiation. The front 
teeth are incisor lik e and behind them are teeth that are rather like 
canines. There are no molars however. The rear teeth are reptil- 
ian cones. 

After flourishing for fifty million years or so, the pelycosaurs 
gave way to a group of synapsids of the order "Therapsida." Un- 
doubtedly, the therapsids were descended from a particular species 
of pelycosaurs. 

The therapsids are clearly further on the road to mammalism 
than any of the pelycosaurs were. The zygomatic arch is much 
more mammal- li ke among them than among the pelycosaurs; so 
much so, in fact, that the feature gives them their name. "Therap- 
sida" is from Greek words meaning "beast opening." In other 
words, the opening in the skull is beastlike where "beast" is the 
common term for what zoologists would call mammals. 

Further, the teeth are much more differentiated among the 
therapsids than among the pelycosaurs. A well-known therapsid 
which lived about 220 million years ago in South Africa had a 
skull and teeth that were so doglike that it is called "Cynogna- 
thus" ("dog jaw"). The back teeth of Cynognathus are even be- 
ginning to look like molars. 

What's more, while the chin of the therapsids was made up of 
seven bones, in typical reptilian fashion, the center bone or "den- 
tary," was by far the largest. The other six bones, three on each 
side, were crowded toward the joint of the lower jaw with the 
upper — on their way to the ear, so to speak. 

In another respect, too, the therapsids showed a "progressive" 
feature. (We tend to call "progressive" anything that see ms to 
move in the direction of ourselves.) Early reptiles, including the 



pelycosaurs, tended to have their legs splayed out so that the up- 
per part, above the knee, was horizontal. This is a rather inef- 
ficient way of suspending the weight of the body. 

Not so the therapsids. In their case, the legs were drawn be- 
neath the body, with the upper parts, as well as the lower, tending 
to be vertical. This makes for better support, allows faster move- 
ment with less energy expenditure, and is a typically mammalian 
characteristic. Apparently, the superior efficiency of the vertical 
leg meant there was no virtue in particularly long toes. Primitive 
reptiles tended to have four or even five joints in their middle 
toes. The therapsids, however, had two joints in the first toes, and 
three joints in the others. Again, this is the way it is in mammals. 

The therapsids, however, did not endure. While we may root 
for them as our great-ever-so-great-grandfathers, the fact is that 
about two hundred million years ago, the "archosaurs," the crea- 
tures representing what we loosely call dinosaurs were coming 
into their own. As they (no ancestors of ours) rapidly grew in 
size and specialization, they crowded out the therapsids. 

By 150 million years ago, the therapsids were clean gone for- 
ever, every single one of them extinct. 

Well, not really! Some small therapsids remained, but they had 
grown so mammal-like, as nearly as we can tell from the very few 
fossil remnants left behind, that we don't call them therapsids 
any more. We call them mammals. 

After the mammals came on the scene, they managed to sur- 
vive through a hundred million years or so of archosaurian 
dominance. Then, after the archosaurs vanished, about seventy 
million years ago, the mammals continued to survive and burst 
into a flood of differentiation and specialization that made this 
latest period of Earth's existence the "age of the mammals." 

The question now is: Why did the mammals survive when the 
therapsids generally did not? The archosaurs proved utterly su- 
perior to the therapsids; why not to those therapsidian offshots, 
the mammals, as well? It couldn't have been that the mammals 
were particularly brainy, because primitive mammals aren't. They 


are not veiy brainy even today, much less a hundred million years 

Nor could it be because of their advanced reproductive system, 
the bearing of live young, for instance. The development of a 
placenta, or even of a pouch, did not take place till near - the end 
of the archosaurian dominance. For nearly a hundred milli on 
year's the mammals survived as egg-layers. 

It couldn't have been their advanced teeth or legs or anything 
skeletal that the therapsids had, generally, for none of that helped 
the therapsids, generally. 

Actually, the best guess is that the trick of survival was warm- 
bloodedness, the development of a constant internal temperature. 
The control of the internal temperature meant that a mammal 
could withstand the direct rays of a hot sun much more easily than 
a reptile could. It meant that a mammal was warm and agile on 
a cold morning when reptiles were cold, stiff, and sluggish. 

If a mammal carefully confined his activity to the chilly hours 
or if he were trapped by a reptile in the heat and could escape by 
darting into the hot sun — it would tend to survive. But for mam- 
mals to have survived in this fashion, their warm-bloodedness must 
have been well developed firm the start and that couldn't happen 

We might conclude, then, that in addition to those changes in 
the therapsids that we can see in the skeleton, there must have 
been additional changes that made warm-bloodedness possible. 
The mammals survived because of all the therapsids, warm- 
bloodedness had developed most efficiently among them. 

Are there any signs of the beginnings of such changes among 
the reptilian precursors of the mammals? Well, a number of spe- 
cies of pelycosaurs had long bony processes to their vertebrae that 
thrust high into the air. Apparently, skin stretched across these 
processes so that pelycosaurs possessed a high, ribbed "sail." 

Why? The American zoologist Alf red S he i wood Romer has sug- 
gested it was an air-conditioning device (like the huge fa nlik e ears 
of the African elephant). Heat is gained or lost through the sur- 
face of the body and the pelycosaurian sail can easily double the 


surface area available. On a cool morning, the sail will pick up 
the Sun's heat and warm the creature much more quickly than 
would be the case for a similar organism without a sail. Again, on a 
hot day, a pelycosaur could stay in the shade and lose heat rap- 
idly through the blood vessels engorging the sail. 

The sail, in short, served to make the pelycosaur's internal 
temperature more nearly constant than was the case in other 
similar reptiles. Their therapsid descendants had no sails, how- 
ever, and it couldn't be that they had abandoned temperature 
control, since their descendants, the mammals, had it in such 
superlative degree. 

It must be that the therapsids had developed something better 
than the sail. A high metabolic rate to produce heat in greater 
quantity might be developed and then hair (which is only modi- 
fied scales) to serve as an insulating device that would cut down 
heat loss on cold days. They might also develop sweat glands to 
get rid of heat on hot days in more efficient manner than by 
means of a sail. 

In short, could the therapsids have been hairy and sweaty, as 
mammals are? We can never tell from the fossils. 

And did those species which best developed hairiness and 
sweatiness become what we call mammals and did they survive 
where the less advanced other therapsids did not? 

Let's look in another direction. In reptiles, the nostrils open 
into the mouth just behind the teeth. This means that reptiles can 
breathe with their mouths closed — and empty. When the mouth 
is full, breathing stops. In the case of the cold-blooded reptiles, 
not much harm is done. The reptilian need for oxygen is relatively 
low and if the supply is cut off temporarily during eating, so what? 

Mammals, however, have to maintain a high metabolic rate at 
all times if they are to be warm-blooded, and that means that the 
oxidation of foodstuffs (from which heat is obtained) must con- 
tinue steadily. The oxygen supply must not be cut off for more 
than a couple of minutes at any time. This is made possible by 
the fact that mammals have a palate, a roof to the mouth. When 
they breathe, air is led above the mouth to the throat. It is only 


when they are actually in the act of swallowing that the breath is 
cut off and this is a matter of a couple of seconds only. 

It is interesting, then, that a number of late therapsid species 
had developed a palate. This might be taken as a pretty good in- 
dication that they were warm-blooded. 

It would seem then that if we could see therapsids in their liv- 
ing state and not as a handful of stony bones, we would see hairy, 
sweaty creatures that we might easily mistake for mammals. We 
might then wonder which hairy, sweaty creatures were reptiles 
and which were mammals. How would we draw the line? 

Nowadays, it might seem, the problem is not a crucial one. All 
the hairy warm-blooded creatures in existence are called mam- 
mals. — And yet, are we justified in doing so? 

In the case of the placentals and the marsupials, we are surely 
justified. They developed their placentas and their pouches about 
eighty million years ago, after the mammals had already existed 
for some hundred million years. The early mammals must have 
been egg-layers and so, therefore, must have been their therapsid 
forebears. If we want to look for the boundary line between 
therapsids and mammals, we must therefore look among the hairy 

As it happens, there are still six species of such hairy egg-layers 
alive today, existing only in Australia, Tasmania, and New 
Guinea, islands that split off from Asia before the more efficient 
placental mammals developed, so that the egg-layers were spared 
what would otherwise have been a fatal competition. The egg- 
layers were first discovered in 1792 and for a while biologists found 
it hai'd to believe they could really exist. It took a long time be- 
fore they got over suspecting a hoax — hairy creatures that laid eggs 
seemed a contradition in terms. 

The best known of the egg-layer's is the "duckbill platypus" (the 
last part of the name means "flat-foot" and the first part refers to 
the homy sheath on its nose that looks like a duck's bill). It is 
also called "Omithorhynchus" from Greek words meaning "bird 



These egg-layers have hair, of course, perfectly good hair, but 
so (veiy likely) had at least some therapsids. The egg-layers also 
produce milk, although their mammary glands have no nipples 
and the young must lick the hah where the milk oozes out. How- 
ever, some therapsid species might also have produced milk in 
that fashion. We can't tell from the bones. 

In some respects, the egg-layers lean strongly toward the side 
of the reptiles. Their body temperature is much less perfectly con- 
trolled than that of other mammals and some of them possess 
venom. The platypus, for instance, has a homy spur at each ankle 
which secretes venom; and though a number of reptiles are 
venomous, no mammals (other than the egg-layers) are. 

Then, too, because they are egg-layers, they have a single ab- 
dominal opening, a "cloaca," which serves as a common passage- 
way for urine, feces, eggs, and sperm. All living birds and reptiles 
(also egg-layers) possess cloacae, but no mammals, other than 
those few egg-layers, do. For this reason, the egg-layers are called 
"monotremes" ("one-hole"). 

To most zoologists, the hair and the milk spell mammal un- 
mistakably, but the eggs, the cloaca, and the venom are sufficiently 
reptilian so that the egg-layers are placed in a subclass "Proto- 
theria" ("first beasts") while all other mammals, marsupials and 
placentals alike, are in the subclass "Theria" ("beasts"). 

The question arises, though: Are the monotremes really the 
first of the mammals, or are they rather the last of the therapsids? 
Are they really reptiles that have the outer appearance of mam- 
mals, as did, perhaps, a number of late therapsid species; or are 
they mammals that have retained some reptilian characteristics? 

This may sound like a purely semantic matter, but zoologists 
must make decisions in such matters and, if possible, come to 
agreement over it. 

An American zoologist, Giles T. MacIntyre, has recently en- 
tered the fray, using skeletal characteristics as the criterion. (We 
have only the skeleton as direct evidence in the therapsid case.) 
He has concentrated on the region near' the ear, where some of 
the reptilian jawbones became mammalian ear bones and where 


you might expect some useful distinction between the two classes. 

There is a "trigeminal nerve" which leads from the jaw 
muscles to the brain. In all reptiles, without exception, it passes 
through a little hole in the skull that lies between two particular 
bones that make up the skull. In all marsupial and placental mam- 
mals, without exception, it passes through a little hole that 
pierces through one of the skull bones. 

Then let us forget about hah and mi lk and eggs and warm- 
bloodedness, and reduce it to a matter of holes in the head. Does 
the trigeminal nerve of the monotremes pass through a skull bone 
or between two skull bones? The answer has been: Through a 
skull bone. 

That would mean the monotremes are mammals. 

Not so, says MacIntyre. The study of the trigeminal nerve was 
made in adult monotremes, where the skull bones are fused and 
the boundaries har'd to make out. In young monotremes, the skull 
bones are not as well developed and are more clearly separated 
(as is true of young mammals generally). In young monotremes, 
MacIntyre says, it is clear' that the trigeminal nerve goes between 
two bones and it is only in the adult skull that bone fusions ob- 
scure the fact. 

If MacIntyre is correct, we may therefore say that the therapsids 
never became entirely extinct and that the monotremes represent 
living therapsids, living reptiles so similar to mammals in some 
ways as to have been considered mammals for nearly two cen- 

Does this matter to anyone but a few zoologists? 

Well, it matters to me. Emotionally, I'm all the way on Mac- 
Intyre's side. I want the therapsids to have survived! 

E — The Problem of History 


In the old days, when I was writing a great deal of fiction, there 
would come, once in a while, moments when I was stymied. Sud- 
denly, I would find I had written myself into a hole and could 
see no way out. To take care of that, I developed a technique 
which invariably worked. 

It was simply this — I went to the movies. Not just any movie. 
I had to pick a movie which was loaded with action but which 
made no demands on the intellect. As I watched, I did my best 
to avoid any conscious thinking concerning my problem, and 
when I came out of the movie I knew exactly what I would have 
to do to put the story back on the track. 

It never failed. 

In fact, when I was working on my doctoral dissertation, too 
many years ago, I suddenly came across a flaw in my logic that I 
had not noticed before and that knocked out everything I had 
done. In utter panic, I made my way to a Bob Hope movie — and 
came out with the necessary change in point of view. 

It is my belief, you see, that thinking is a double phenomenon, 
like breathing. 

You can control breathing by deliberate voluntary action: you 
can breathe deeply and quickly, or you can hold your breath al- 
together, regardless of the body's needs at the time. This, how- 
ever, doesn't work well for very long. Your chest muscles grow 
tired, your body clamors for more oxygen, or less, and you relax. 
The automatic involuntary control of breathing takes over, ad- 
justs it to the body's needs, and unless you have some respiratory 
disorder, you can forget about the whole thing. 

Well, you can think by deliberate voluntary action, too, and I 
don't think it is much more efficient on the whole than voluntary 

1 6 o 


breath control is. You can deliberately force your mind through 
channels of deductions and associations in search of a solution 
to some problem and before long you have dug mental furrows 
for yourself and find yourself circling round and round the same 
limited pathways. If those pathways yield no solution, no amount 
of further conscious thought will help. 

On the other hand, if you let go, then the thinking process 
comes under automatic involuntary control and is more apt to 
take new pathways and make erratic associations you would not 
think of consciously. The solution will then come while you think 
you are not thinking. 

The trouble is, though, that conscious thought involves no 
muscular action and so there is no sensation of physical weari- 
ness that would force you to quit. What's more, the panic of neces- 
sity tends to force you to go on uselessly, with each added bit of 
useless effort adding to the panic in a vicious cycle. 

It is my feeling that it helps to relax, deliberately, by subjecting 
your mind to material complicated enough to occupy the volun- 
tary faculty of thought, but superficial enough not to engage 
the deeper involuntary one. In my case, it is an action movie; in 
your case, it might be something else. 

I suspect it is the involuntary faculty of thought that gives rise 
to what we call "a flash of intuition," something that I imagine 
must be merely the result of unnoticed thinking. 

Perhaps the most famous flash of intuition in the history of 
science took place in the city of Syracuse in third-century B.C. 
Sicily. Bear with me and I will tell you the story — 

About 250 B.C, the city of Syracuse was experiencing a kind of 
Golden Age. It was under the protection of the rising power 
of Rome, but it retained a king of its own and considerable self- 
government; it was prosperous; and it had a flourishing intellec- 
tual life. 

The king was Hieron II, and he had commissioned a new 
golden crown from a goldsmith, to whom he had given an ingot of 
gold as raw material. Hieron, being a practical man, had carefully 


weighed the ingot and then weighed the crown he received back. 
The two weights were precisely equal. Good deal! 

But then he sat and thought for a while. Suppose the goldsmith 
had subtracted a little bit of the gold, not too much, and had 
substituted an equal weight of the considerably less valuable 
copper. The resulting alloy would still have the appearance of 
pure gold, but the goldsmith would be plus a quantity of gold 
over and above his fee. He would be buying gold with copper, so 
to speak, and Hieron would be neatly cheated. 

Hieron didn't like the thought of being cheated any more than 
you or I would, but he didn't know how to find out for sure if 
he had been. He could scarcely punish the goldsmith on mere 
suspicion. What to do? 

Fortunately, Hieron had an advantage few rulers in the history 
of the world could boast. He had a relative of considerable talent. 
The relative was named Archimedes and he probably had the 
greatest intellect the world was to see prior to the birth of 

Archimedes was called in and was posed the problem. He had 
to determine whether the crown Hieron showed him was pure 
gold, or was gold to which a small but significant quantity of 
copper had been added. 

If we were to reconstruct Archimedes' reasoning, it might go 
as follows. Gold was the densest known substance (at that time). 
Its density in modem terms is 19.3 grams per cubic centimeter. 
This means that a given weight of gold takes up less volume 
than the same weight of anything else! In fact, a given weight of 
pure gold takes up less volume than the same weight of any 
kind of impure gold. 

The density of copper is 8.92 grams per cubic centimeter, just 
about half that of gold. If we consider 100 grams of pure gold, for 
instance, it is easy to calculate it to have a volume of 5.18 cubic 
centimeters. But suppose the 100 grams of what looked like pure 
gold was really only 90 grams of gold and 10 grams of copper. 
The 90 grams of gold would have a volume of 4.66 cubic centi- 


meters, while the 10 grams of copper would have a volume of 1.12 
cubic centimeters; for a total value of 5.78 cubic centimeters. 

The difference between 5.18 cubic centimeters and 5.78 cubic 
centimeters is quite a noticeable one, and would instantly tell if 
the crown were of pure gold, or if it contained 10 per cent copper 
(with the missing 10 per cent of gold tucked neatly in the gold- 
smith's strongbox). 

All one had to do, then, was measure the volume of the crown 
and compare it with the volume of the same weight of pure gold. 

The mathematics of the time made it easy to measure the 
volume of many simple shapes: a cube, a sphere, a cone, a cylinder, 
any flattened object of simple regular shape and known thick- 
ness, and so on. 

We can imagine Archimedes saying, "All that is necessary, sire, 
is to pound that crown flat, shape it into a square of uniform 
thickness, and then I can have the answer for you in a moment." 

Whereupon Hieron must certainly have snatched the crown 
away and said, "No such thing. I can do that much without you; 
I've studied the principles of mathematics, too. This crown is a 
highly satisfactory work of art and I won't have it damaged. Just 
calculate its volume without in any way altering it." 

But Greek mathematics had no way of determining the volume 
of anything with a shape as irregular as the crown, since integral 
calculus had not yet been invented (and wouldn't be for two 
thousand years, almost). Archimedes would have had to say, 
"There is no known way, sire, to carry through a non-destructive 
determination of volume." 

"Then think of one," said Hieron testily. 

And Archimedes must have set about thinking of one, and got- 
ten nowhere. Nobody knows how long he thought, or how hard, 
or what hypotheses he considered and discarded, or any of the 

What we do know is that, worn out with thinking, Archimedes 
decided to visit the public baths and relax. I think we are quite 
safe in saying that Archimedes had no intention of taking his 
problem to the baths with him. It would be ridiculous to imagine 


he would, for the public baths of a Greek metropolis weren't 
intended for that sort of thing. 

The Greek baths were a place for relaxation. Half the social 
aristocracy of the town would be there and there was a great 
deal more to do than wash. One steamed one's self, got a mas- 
sage, exercised, and engaged in general socializing. We can be sure 
that Archimedes intended to forget the stupid crown for a while. 

One can envisage him engaging in light talk, discussing the 
latest news horn Alexandria and Carthage, the latest scandals in 
town, the latest funny jokes at the expense of the country-squire 
Romans — and then he lowered himself into a nice hot bath which 
some bumbling attendant had filled too full. 

The water in the bath slopped over as Archimedes got in. Did 
Arc him edes notice that at once, or did he sigh, sink back, and 
paddle his feet awhile before noting the water-slop. 1 guess the 
latter. But, whether soon or late, he noticed, and that one fact, 
added to all the chains of reasoning his brain had been working 
on during the period of relaxation when it was unhampered 
by the comparative stupidities (even in Archimedes) of voluntary 
thought, gave Archimedes his answer in one blinding flash of 

Jumping out of the bath, he proceeded to run home at top 
speed through the streets of Syracuse. He did not bother to put 
on his clothes. The thought of Archimedes running naked through 
Syracuse has titillated dozens of generations of youngsters who 
have heard this story, but 1 must explain that the ancient Greeks 
were quite lighthearted in their attitude toward nudity. They 
thought no more of seeing a naked man on the streets of Syra- 
cuse, than we would on the Broadway stage. 

And as he ran, Archimedes shouted over and over, "I've got it! 
I've got it!" Of course, knowing no English, he was compelled to 
shout it in Greek, so it came out, "Eureka! Eureka!" 

Archimedes' solution was so simple that anyone could under- 
stand it — once Archimedes explained it. 

If an object that is not affected by water in any way, is im- 
mersed in water, it is bound to displace an amount of water 



equal to its own volume, since two objects cannot occupy the 
same space at the same time. 

Suppose, then, you had a vessel large enough to hold the crown 
and suppose it had a small overflow spout set into the middle of its 
side. And suppose further that the vessel was fill ed with water 
exactly to the spout, so that if the water level were raised a bit 
higher, however slightly, some would overflow. 

Next, suppose that you carefully lower the crown into the 
water. The water level would rise by an amount equal to the 
volume of the crown, and that volume of water would pour out 
the overflow and be caught in a small vessel. Next, a lump of gold, 
known to be pure and exactly equal in weight to the crown, is 
also immersed in the water and again the level rises and the over- 
flow is caught in a second vessel. 

If the crown were pure gold, the overflow would be exactly the 
same in each case, and the volumes of water caught in the two 
small vessels would be equal. If, however, the crown were of alloy, 
it would produce a larger overflow than the pure gold would and 
this would be easily noticeable. 

What's more, the crown would in no way be harmed, defaced, 
or even as much as scratched. More important, Archimedes had 
discovered the "principle of buoyancy." 

And was the crown pure gold? I've heard that it turned out to 
be alloy and that the goldsmith was executed, but I wouldn't 
swear to it. 

How often does this "Eureka phenomenon" happen? How of- 
ten is there this flash of deep insight during a moment of relaxa- 
tion, this triumphant ay of "I've got it! I've got it!" which must 
surely be a moment of the purest ecstasy this sony world can 

I wish there were some way we could tell. I suspect that in the 
history of science it happens often; I suspect that veiy few signifi- 
cant discoveries are made by the pure technique of voluntary 
thought; I suspect that voluntary thought may possibly prepare 


the ground (if even that), but that the final touch, the real in- 
spiration, comes when thinking is under involuntary control. 

But the world is in a conspiracy to hide that fact. Scientists are 
wedded to reason, to the meticulous working out of consequences 
from assumptions, to the careful organization of experiments 
designed to check those consequences. If a certain line of experi- 
ments ends nowhere, it is omitted from the final report. If an 
inspired guess turns out to be correct, it is not reported as an 
inspired guess. Instead, a solid line of voluntary thought is in- 
vented after the fact to lead up to the thought, and that is what 
is inserted in the final report. 

The result is that anyone reading scientific papers would swear 
that nothing took place but voluntary thought maintaining a 
steady clumping stride from origin to destination, and that just 
can't be true. 

It's such a shame. Not only does it deprive science of much of 
its glamour (how much of the dramatic story in Watson's Double 
Helix do you suppose got into the final reports announcing the 
great discovery of the structure of DNA?*), but it hands over 
the important process of "insight," "inspiration," "revelation" to 
the mystic. 

The scientist actually becomes ashamed of having what we 
might call a revelation, as though to have one is to betray reason 
— when actually what we call revelation in a man who has devoted 
his life to reasoned thought, is after all merely reasoned thought 
that is not under voluntary control. 

Only once in a while in modern times do we ever get a glimpse 
into the workings of involuntary reasoning, and when we do, it 
is always fascinating. Consider, for instance, the case of Friedrich 
August Kekule von Stradonitz. 

In Kekule's time, a century and a quarter ago, a subject of great 
interest to chemists was the structure of organic molecules (those 
associated with living tissue). Inorganic molecules were generally 
simple in the sense that they were made up of few atoms. Water 

* I'll tell you, in case you're curious. None! 



molecules, for instance, are made up of two atoms of hydrogen 
and one of oxygen (H 2 0). Molecules of ordinary salt are made 
up of one atom of sodium and one of chlorine (NaCl), and so on. 

Organic molecules, on the other hand, often contained a large 
number of atoms. Ethyl alcohol molecules have two carbon atoms, 
six hydrogen atoms, and an oxygen atom (C 2 H 6 0 ); the molecule 
of ordinary cane sugar is C12H22011, and other molecules are 
even more complex. 

Then, too, it is sufficient, in the case of inorganic molecules 
generally, merely to know the kinds and numbers of atoms in 
the molecule; in organic molecules, more is necessary. Thus, 
dimethyl ether has the formula C2H e O, just as ethyl alcohol does, 
and yet the two are quite different in properties. Apparently, the 
atoms are arranged differently within the molecules — but how to 
determine the arrangements? 

In 1852, an English chemist, Edward Frankland, had noticed 
that the atoms of a particular' element tended to combine with a 
fixed number of other atoms. This combining number was called 
"valence." Kekule in 1858 reduced this notion to a system. The 
carrion atom, he decided (on the basis of plenty of chemical evi- 
dence) had a valence of four" the hydrogen atom, a valence of 
one; and the oxygen atom, a valence of two (and so on). 

Why not represent the atoms as their symbols plus a number 
of attached dashes, that number being equal to the valence. Such 
atoms could then be put together as though they were so many 
Tinker Toy units and "structural formulas" could be built up. 

It was possible to reason out that the structural formula 
H H 

of ethyl alcohol was H — C — C — 0 — H > while that of dimethyl 

I I 
H H 

H H 

1 1 

ether was H — C — 0 — C — H . 

I I 

H H 



In each case, there were two carbon atoms, each with four 
dashes attached; six hydrogen atoms, each with one dash attached; 
and an oxygen atom with two dashes attached. The molecules 
were built up of the same components, but in different arrange- 

Kekule's theory worked beautifully. It has been immensely 
deepened and elaborated since his day, but you can still find struc- 
tures veiy much like Kekule's Tinker Toy formulas in any modem 
chemical textbook. They represent oversimplifications of the true 
situation, but they remain extremely useful in practice even so. 

The Kekule structures were applied to many organic molecules 
in the years after 1858 and the similarities and contrasts in the 
structures neatly matched similarities and contrasts in properties. 
The key to the rationalization of organic chemistry had, it seemed, 
been found. 

Yet there was one disturbing fact. The well-known chemical 
benzene wouldn't fit. It was known to have a molecule made up 
of equal numbers of carbon and hydrogen atoms. Its molecular 
weight was known to be 78 and a single carbon-hydrogen combi- 
nation had a weight of 13. Therefore, the benzene molecule had 
to contain six carbon-hydrogen combinations and its formula had 
to be C 6 H 8 . 

But that meant trouble. By the Kekule formulas, the hydro- 
carbons (molecules made up of carbon and hydrogen atoms only) 
could easily be envisioned as chains of carbon atoms with hydro- 
gen atoms attached. If all the valences of the carbon atoms were 
filled with hydrogen atoms, as in "hexane," whose molecule 
looks like this — 

I I I I I I 

H H H H H H 

the compound is said to be saturated. Such saturated hydrocarbons 
were found to have very little tendency to react with other sub- 



If some of the valences were not filled, unused bonds were 
added to those connecting the carbon atoms. Double bonds were 
formed as in "hexene" — 

H H H H H H 

I I I I I I 

H-C-C-C= C-C-C-H 


H H H H 

Hexene is unsaturated, for that double bond has a tendency to 
open up and add other atoms. Hexene is chemically active. 

When six carbons are present in a molecule, it takes fourteen 
hydrogen atoms to occupy all the valence bonds and make it inert 
— as in hexane. In hexene, on the other hand, there are only twelve 
hydrogens. If there were still fewer hydrogen atoms, there would 
be more than one double bond; there might even be triple bonds, 
and the compound would be still more active than hexene. 

Yet benzene, which is C 6 H 6 and has eight fewer hydrogen 
atoms than hexane, is less active than hexene, which has only two 
fewer hydrogen atoms than hexane. In fact, benzene is even less 
active than hexane itself. The six hydrogen atoms in the benzene 
molecule seem to satisfy the six carbon atoms to a greater extent 
than do the fourteen hydrogen atoms in hexane. 

For heaven's sake, why? 

This might seem unimportant. The Kekule formulas were so 
beautifully suitable in the case of so many compounds that one 
might simply dismiss benzene as an exception to the general rale. 

Science, however, is not English grammar. You can't just 
categorize something as an exception. If the exception doesn't fit 
into the general system, then the general system must be wrong. 

Or, take the more positive approach. An exception can often 
be made to fit into a general system, provided the general system 
is broadened. Such broadening generally represents a great advance 
and for this reason, exceptions ought to be paid great attention. 

For some seven years, Kekule faced the problem of benzene and 
tried to puzzle out how a chain of six carbon atoms could be com- 



pletely satisfied with as few as six hydrogen atoms in benzene 
and yet be left unsatisfied with twelve hydrogen atoms in hexene. 

Nothing came to him! 

And then one day in 1865 ( ne tells the story himself) he was 
in Ghent, Belgium, and in order to get to some destination, he 
boarded a public bus. He was tired and, undoubtedly, the droning 
beat of the horses' hooves on the cobblestones, lulled him. He fell 
into a comatose half-sleep. 

In that sleep, he seemed to see a vision of atoms attaching 
themselves to each other in chains that moved about. (Why not? 
It was the sort of thing that constantly occupied his waking 
thoughts.) But then one chain twisted in such a way that head 
and tail joined, forming a ring — and Kekule woke with a start. 

To himself, he must surely have shouted "Eureka," for indeed he 
had it. The six carbon atoms of benzene formed a ring and not a 
chain, so that the structural formula looked like this: 



« I 



To be sure, there were still three double bonds, so you might 
think the molecule had to be very active — but now there was a 
difference. Atoms in a ring might be expected to have different 
properties from those in a chain and double bonds in one case 
might not have the properties of those in the other. At least, 
chemists could work on that assumption and see if it involved 
them in contradictions. 

It didn't. The assumption worked excellently well. It turned out 
that organic molecules could be divided into two groups: aromatic 
and aliphatic. The former had the benzene ring (or certain other 
similar rings) as part of the structure and the latter did not. Al- 


lowing for different properties within each group, the Kekule 
structures worked very well. 

For nearly seventy years, Kekule's vision held good in the hard 
field of actual chemical techniques, guiding the chemist through 
the jungle of reactions that led to the synthesis of more and 
more molecules. Then, in 1932, Linus Pauling applied quantum 
mechanics to chemical structure with sufficient subtlety to explain 
just why the benzene ring was so special and what had proven 
correct in practice proved correct in theory as well. 

Other cases? Certainly. 

In 1764, the Scottish engineer James Watt was working as 
an instrument maker for the University of Glasgow. The univer- 
sity gave him a model of a Newcomen steam engine, which 
didn't work well, and asked him to fix it. Watt fixed it without 
trouble, but even when it worked perfectly, it didn't work well. It 
was far too inefficient and consumed incredible quantities of 
fuel. Was there a way to improve that? 

Thought didn't help; but a peaceful, relaxed walk on a Sunday 
afternoon did. W att returned with the key notion in mind of using 
two separate chambers, one for steam only and one for cold water 
only, so that the same chamber did not have to be constantly 
cooled and reheated to the infinite waste of fuel. 

The Irish mathematician William Rowan Hamilton worked up 
a theory of "quaternions" in 1843 Dut couldn't complete that 
theory until he grasped the fact that there were conditions under 
which p X q was not equal to q X p. The necessary thought came 
to him in a flash one time when he was walking to town with his 

The German physiologist Otto Loewi was working on the 
mechanism of nerve action, in particular, on the chemicals pro- 
duced by nerve endings. He woke at 3 A.M. one night in 1921 with 
a perfectly clear notion of the type of experiment he would have 
to run to settle a key point that was puzzling him. He wrote it 
down and went back to sleep. When he woke in the morning, he 
found he couldn't remember what his inspiration had been. He 


remembered he had written it down, but he couldn't read his 

The next night, he woke again at 3 A.M. with the clear thought 
once more in mind. This time, he didn't fool around. He got up, 
dressed himself, went straight to the laboratory and began work. 
By 5 A.M. he had proved his point and the consequences of his 
findings became important enough in later years so that in 1936 
he received a share in the Nobel prize in medicine and physiology. 

How very often this sort of thing must happen, and what a 
shame that scientists are so devoted to their belief in conscious 
thought that they so consistently obscure the actual methods by 
which they obtain their results. 

14 - 


Rationalists have a hard time of it, because the popular view is 
that they are committed to "explaining" everything. 

This is not so. Rationalists maintain that the proper way of 
arriving at an explanation is through reason — but there is no 
guarantee that some particular phenomenon can be explained in 
that fashion at some given moment in history or from some 
given quantity of observation.* 

Yet how often I (or any rationalist) am presented with some- 
thing odd and am challenged, "How do you explain that?" The 
implication is that if I don't explain it instantly to the satisfac- 
tion of the individual posing the question, then the entire struc- 
ture of science may be considered to be demolished. 

But things happen to me, too. One day in April 1967, my car' 
broke down and had to be towed to a garage. In seventeen years 
of driving various cars, that was the first time I ever had to en- 
dure the humiliation of being towed. 

When do you suppose the second time was? — Two hour's later, 
on the same day, for a completely different reason. 

Seventeen years without a tow, and then two tows on the same 
day! And how do you explain that. Dr. Asimov? (Gremlins? 
A vengeful Deity? An extraterrestrial conspiracy?) 

On the second occasion, I did indeed loudly advance all three 
theories to my unruffled garageman. His theory (he was also a 
rationalist) was that my car' was old enough to be falling apart. 
So I bought a new car'. 

Let's look at it this way! To every single person on Earth, a 
large number of events, great, small, and insignificant, happen 

* It is the mystics, really, who are committed to explaining everything, for 
they need nothing but imagination and words — any words, chosen at random. 


each day. Every one of those events has some probability of oc- 
currence, though we can't always decide the exact probability in 
each case. On the average, though, we might imagine that one 
out of every thousand events has an only one-in-a-thousand 
chance of happening; one out of every million events has an only 
one-in-a-million chance of happening; and so on. 

This means that every one of us is constantly experiencing some 
pretty low-probability events. That is the normal result of chance. 
If any of us went an appreciable length of time with nothing un- 
usual happening, that would be very unusual. 

And suppose we don't restrict ourselves to one person, but con- 
sider, instead, all the lives that have ever been lived. The number 
of events then increases by a factor of some sixty billion and we 
can assume that sometime, to someone, something will happen 
that is sixty billion times as improbable as anything happening 
to some other particular man. Even such an event requires no 
explanation. It is part of our normal universe going along its busi- 
ness in a normal way. 

Examples? We've all heard very odd coincidences that have 
happened to someone's second cousin, odd things that represent 
such an unusual concatenation of circumstance that surely we 
must admit the existence of telepathy or flying saucers or Satan 
or something. 

Let me offer something, too. Not something that happened to 
my second cousin, but to a notable figure of the past whose life 
is quite well documented. Something very unusual happened to 
him, which in all my various and miscellaneous reading of history 
I have never seen specifically pointed out. I will, therefore, stress 
it to you as something more unusual and amazing than anything 
I have ever come across, and even so, it still doesn't shake my be- 
lief in the supremacy of the rational view of the universe. Here 
goes — 

The man in question was Gnaeus Pompeius, who is better 
known to English-speaking individuals as Pompey. 

Pompey was born in 106 B.C. and the first forty -two years of his 




life were characterized by uniform good fortune. Oh, I dare say 
he stubbed his toe now and then and got attacks of indigestion 
at inconvenient times and lost money on the gladiatorial con- 
tests — but in the major aspects of life, he remained always on the 
winning side. 

Pompey was born at a time when Rome was torn by civil war 
and social turmoil. The Italian allies, who were not Roman citi- 
zens, rose in rebellion against a Roman aristocracy who wouldn't 
extend the franchise. The lower classes, who were feeling the 
pinch of a tightening economy, now that Rome had completed 
the looting of most of the Mediterranean area, were struggling 
against the senators, who had kept most of the loot. 

When Pompey was in his teens, his father was trying to walk 
the tightrope. The elder Pompey had been a general who had 
served as consul in 89 B.C, and had defeated the Italian non- 
citizens and celebrated a triumph. But he was not an aristocrat 
by birth and he tried to make a deal with the radicals. This might 
have gotten him in real trouble, for he had worked himself into 
a spot where neither side trusted him, but in 87 B.C. he died in the 
course of an epidemic that swept his army. 

That left young Pompey as a fatherless nineteen-year-old who 
had inherited enemies on both sides of the civil war. 

He had to choose and he had to choose carefully. The radicals 
were in control of Rome, but off in Asia Minor, fighting a war 
against Rome's enemies, was the reactionary general Lucius Cor- 
nelius Sulla. 

Pompey, uncertain as to which side would win, lay low and out 
of sight. When he heard that Sulla was returning, victorious, from 
Asia Minor, he made his decision. He chose Sulla as probable 
victor. At once, he scrabbled together an army from among those 
soldiers who had fought for his father, loudly proclaimed himself 
on Sulla's side, and took the field against the radicals. 

There was his first stroke of fortune. He had backed the right 
man. Sulla arrived in Italy in 83 B.C. and began winning at once. 
By 82 B.C. he had wiped out the last opposition in Italy and at 
once made himself dictator. For three years he was absolute ruler 


of Rome. He reorganized the government and placed the senato- 
rial aristocrats firmly in control. 

Pompey benefited, for Sulla was properly grateful to him. Sulla 
sent Pompey to Sicily, then to Africa, to wipe out the disorganized 
forces that still clung to the radical side there, and this was done 
without trouble. 

The victories were cheap and Pompey's troops were so pleased 
that they acclaimed Pompey as "the Great," so that he became 
Gnaeus Pompeius Magnus — the only Roman to bear this utterly 
un-Roman cognomen. Later accounts say that he received this 
name because of a striking physical resemblance between himself 
and Alexander the Great, but such a resemblance could have ex- 
isted only in Pompey's own imagination. 

Sulla ordered Pompey to disband his army after his African 
victories but Pompey refused to do so, preferring to stay sur- 
rounded by his loyal men. Ordinarily, one did not lightly cross 
Sulla, who had no compunctions whatever about ordering a few 
dozen executions before breakfast. Pompey, however, proceeded 
to marry Sulla's daughter. Apparently, this won Sulla over to the 
point of not only accepting the title of "the Great" for the young 
man, but also to the point of allowing him to celebrate a triumph 
in 79 B.C. even though he was below the minimum age at which 
triumphs were permitted. 

Almost immediately thereafter, Sulla resigned the dictatorship, 
feeling his work was done, but Pompey's career never as much as 
stumbled. He now had a considerable reputation (based on his 
easy victories). What's more, he was greedy for further easy vic- 

For instance, after Sulla's death, a Roman general, Marcus 
Aemilius Lepidus, turned against Sulla's policies. The reactionary 
Senate at once sent an army against him. The senatorial army 
was led by Quintus Catulus, with Pompey as second-in-command. 
Until then, Pompey had supported Lepidus, but again he guessed 
the winning side in time. Catulus easily defeated Lepidus, and 
Pompey managed to get most of the credit. 

There was trouble in Spain at this time, for it was the last 


stronghold of radicalism. In Spain, a radical general, Quintus 
Sertorius, maintained him self. Under him, Spain was virtually 
independent of Rome and was blessed with an enlightened gov- 
ernment, for Sertorius was an efficient and liberal administrator. 
He treated the native Spaniards well, set up a Senate into which 
they were admitted, and established schools where their young 
men were trained in Roman style. 

Naturally, the Spaniards, who for some centuries had had a 
reputation as fierce and resolute warriors, fought heart and soul 
on the side of Sertorius. When Sulla sent Roman armies into 
Spain, they were defeated. 

So, in 77 B.C, Pompey, all in a glow over Catulus' easy victoiy 
over Lepidus, offered to go to Spain to take care of Sertorius. 
The Senate was willing and off to Spain marched Pompey and 
his army. On his way through Gaul, he found the dispirited rem- 
nants of Lepidus' old army. Lepidus himself was dead by now but 
what was left of his men were under Marcus Brutus (whose son 
would, one day, be a famous assassin). 

There was no trouble in handling the broken army and Pompey 
offered Brutus his life if he would surrender. Brutus surrendered 
and Pompey promptly had him executed. One more easy victoiy, 
topped by treachery, and Pompey's reputation increased. 

On to Spain went Pompey. In Spain, a sturdy old Roman gen- 
eral, Metellus Pius, was unsuccessfully hying to cope with Serto- 
rius. Vaingloriously, Pompey advanced on his own to take over the 
job — and Sertorius, who was the first good general Pompey had 
yet encountered, promptly gave the young man a first-class drub- 
bing. Pompey's reputation might have withered then and there, 
but just in time, Metellus approached with reinforcements and 
Sertorius had to withdraw. At once, Pompey called it a victoiy, 
and, of course, got the credit for it. His luck held. 

For five years, Pompey remained in Spain, hying to handle 
Sertorius, and for five years he failed. And then he had a stroke 
of luck, the luck that never failed Pompey, for Sertorius was as- 
sassinated. With Sertorius gone, the resistance movement in Spain 
collapsed. Pompey could at once win another of his easy victories 


and could then return to Rome in 71 B.C., claiming to have 
cleaned up the Spanish mess. 

But couldn't Rome have seen it took him five years? 

No, Rome couldn't, for all the time Pompey had been in Spain, 
Italy itself had been going through a terrible time and there had 
been no chance of keeping an eye on Spain. 

A band of gladiators, under Spartacus, had revolted. Many dis- 
possessed flocked to Spartacus' side and for two years, Spartacus 
(a skillful fighter) destroyed every Roman army sent out against 
him and struck terror into the heart of every aristocrat. At the 
height of his power he had 90,000 men under his command and 
controlled almost all of southern Italy. 

In 72 B.C., Spartacus fought his way northward to the Alps, 
intending to leave Italy and gain permanent freedom in the bar- 
barian regions to the north. His men, however, misled by their 
initial victories, preferred to remain in Italy in reach of more loot. 
Spartacus turned south again. 

The senators now placed an army under Marcus Licinius Cras- 
sus, Rome's richest and most crooked businessman. In two battles, 
Crassus managed to defeat the gladiatorial army and in the second 
one, Spartacus was killed. Then, just as Crassus had finished the 
hard work, Pompey returned with his Spanish army and hastily 
swept up the demoralized remnants. He immediately represented 
himself, successfully, as the man who had cleaned up the gladia- 
torial mess after having taken care of Spain. The result was that 
Pompey was allowed to celebrate a triumph, but poor Crassus 

The Senate, though, was growing nervous. They were not sure 
they trusted Pompey. He had won too many victories and was 
becoming entirely too popular. 

Nor did they like Crassus (no one did). For all his wealth, 
Crassus was not a member of the aristocratic families and he grew 
angry at being snubbed by the socially superior Senate. Crassus 
began to court favor with the people with well-placed philan- 
thropies. He also began to court Pompey. 

Pompey always responded to courting and, besides, had an un- 



failing nose for the winning side. He and Crassus ran for the 
consulate in 70 B.C. (two consuls were elected each year), and 
they won. Once consul, Crassus began to undo Sulla's reforms of 
a decade earlier in order to weaken the hold of the senatorial aris- 
tocracy on the government. Pompey, who had been heart and 
soul with Sulla when that had been the politic thing to do, turned 
about and went along with Crassus, though not always happily. 

But Rome was still in trouble. The West had been entirely paci- 
fied, but there was mischief at sea. Roman conquests had broken 
down the older stable governments in the East without having, as 
yet, established anything quite as stable in their place. The result 
was that piracy was rife throughout the eastern Mediterranean. 
It was a rare ship that could get through safely and, in particular, 
the grain supply to Rome itself had become so precarious that 
the price of food skyrocketed. 

Roman attempts to clear out the pirates failed, partly because 
the generals sent to do the job were never given enough power. 
In 67 B.C, Pompey maneuvered to have hi ms elf appointed to the 
task — but under favorable conditions. The Senate, in a panic over 
the food supply, leaped at the bait. 

Pompey was given dictatorial powers over the entire Mediter- 
ranean coast to a distance of fifty miles inland for three years and 
was told to use that time and the entire Roman fleet to destroy 
the pirates. So great was Roman confidence in Pompey that food 
prices fell as soon as news of his appointment was made public. 

Pompey was lucky enough to have what no previous Roman 
had — adequate forces and adequate power. Nevertheless one must 
admit that he did well. In three months, not three years, he 
scoured the Mediterranean clear of piracy. 

If he had been popular before, he was Rome's hero now. 

The only place where Rome still faced trouble was in eastern 
Asia Minor, where the kingdom of Pontus had been fighting 
Rome with varying success for over twenty years. It had been 
against Pontus that Sulla had won victories in the East, yet Pontus 
kept fighting on. Now a Roman general, Lucius Licinius Lucullus, 


had almost finished the job, but he was a hard-driving martinet, 
hated by his soldiers. 

When Lucullus' army began to mutiny in 66 B.C., just when one 
more drive would finish Pontus, he was recalled and good old 
Pompey was sent castwaid to replace him. Pompey's reputation 
preceded him; Lucullus' men cheered him madly and for him did 
what they wouldn't do for Lucullus. They marched against Pontus 
and beat it. Pompey supplied the one last push and, as always, 
demanded and accepted credit for the whole thing. 

All of Asia Minor was now either Roman outright or was under 
the control of Roman puppet governments. Pompey therefore 
decided to clean up the East altogether. He marched southward 
and around Antioch found the last remnant of the Seleucid Em- 
pire, established after the death of Alexander the Great two and 
a half centuries before. It was now ruled by a nonentity called An- 
tiochus XIII. Pompey deposed him, and annexed the empire to 
Rome as the province of Syria. 

Still further south was the kingdom of Judea. It had been in- 
dependent for less than a century, under the rule of a line of kings 
of the Maccabean family. Two of the Maccabeans were now fight- 
ing over the throne and one appealed to Pompey. 

Pompey at once marched into Judea and laid siege to Jerusalem. 
Ordinarily, Jerusalem was a hard nut to crack, for it was built on 
a rocky prominence with a reliable water supply; it had good walls; 
and it was usually defended with fanatic vigor. 

Pompey, however, noticed that every seven days things were 
quiet. Someone explained to him that on the Sabbath, the Jews 
wouldn't fight unless attacked and even then fought without real 
conviction. It must have taken quite a while to convince Pompey 
of such a ridiculous thing but, once convinced, he used a few Sab- 
baths to bring up his siege machinery without interference, and 
finally attacked on another Sabbath. No problem. 

Pompey ended the Maccabean kingdom and annexed Judea to 
Rome while allowing the Jews to keep their religious freedom, 
their Temple, their high-priests, and their peculiar', but useful. 


Pompey was foity-two years old at this time, and success had 
smiled at him without interruption. 1 now skip a single small event 
in Pompey 's life and represent it by a line of asterisks: one ap- 
parently unimportant circumstance. 

Pompey returned to Italy in 61 B.C. absolutely on top of the 
world, boasting (with considerable exaggeration) that what he 
had found as the eastern border of the realm he had left at its 
center. He received the most magnificent triumph Rome had ever 
seen up to that time. 

The Senate was in terror lest Pompey make hi ms elf a dictator 
and turn to the radicals. This Pompey did not do. Once, twenty 
years before, when he had an army, he kept that army even at the 
risk of Sulla's displeasure. Now, something impelled him to give 
up his army, disband it, and assume a role as a private citizen. 
Perhaps he was convinced that he had reached a point where the 
sheer magic of his name would allow him to dominate the re- 

At last, though, his nose for the right action failed him. And 
once having failed him, it failed him forever after. 

To begin with, Pompey asked the Senate to approve everything 
he had done in the East, his victories, his treaties, his depositions 
of kings, his establishment of provinces. He also asked the Senate 
to distribute land to his soldiers, for he himself had promised 
them land. He was sure that he had but to ask and he would be 

Not at all. Pompey was now a man without an army and the 
Senate insisted on considering each individual act separately and 
nit-pickingly. As for land grants, that was rejected. 

What's more, Pompey found that he had no one on his side 
within the government. All his vast popularity suddenly seemed 
to count for nothing as all parties turned against him for no dis- 
cernible reason. What's more, Pompey could do nothing about it. 
Something had happened, and he was no longer the clever, 


golden-boy Pompey he had been before 64 B.C. Now he was un- 
certain, vacillating, and weak. 

Even Crassus was no longer his friend. Crassus had found some- 
one else: a handsome, charming individual with a silver tongue 
and a genius for intrigue — a man named Julius Caesar. Caesar 
was a playboy aristocrat but Crassus paid off the young man's 
enormous debts and Caesar scr\'cd him well in return. 

While Pompey was struggling with the Senate, Caesar was off in 
Spain, winning some small victories against rebellious tribes and 
gathering enough ill-gotten wealth (as Roman generals usually 
did) to pay off Crassus and make himself independent. When 
he returned to Italy and found Pompey furious with the Senate, 
he arranged a kind of treaty of alliance between himself, Crassus, 
and Pompey — the "First Triumvirate." 

But it was Caesar and not Pompey who profited from this. It 
was Caesar who used the alliance to get hi ms elf elected consul 
in 59 B.C. Once consul, Caesar' controlled the Senate with almost 
contemptuous ease, driving the other consul, a reactionary, into 
house arrest. 

One thing Caesar' did was to force the aristocrats of the Senate 
to grant all of Pompey's demands. Pompey got the ratification of 
all of his acts and he got the land for his soldier's — and yet he did 
not profit from this. Indeed, he suffered humiliation, for it was 
quite clear' that he was standing, hat in hand, while Caesar' gra- 
ciously bestowed largesse on him. 

Yet Pompey could do nothing, for he had married Juha, Cae- 
sar's daughter. She was beautiful and winning and Pompey was 
crazy about her. While he had her, he could do nothing to cross 

Caesar' was running everything now. In 58 B.C., he suggested 
that he, Pompey, and Crassus each have a province in which they 
could win military victories. Pompey was to have Spain; Crassus 
was to have Syria; and Caesar' was to have southern Gaul, which 
was then in Roman hands. Each was to be in charge for five years. 

Pompey was delighted. In Syria, Crassus would have to face 
the redoubtable Parthian kingdom, and in Gaul, Caesar' would 


have to face the fierce-fighting barbarians of the North. With luck, 
both would end in disaster, since neither was a trained military 
man. As for Pompey, since Spain was quiet, he could stay in Italy 
and control the government. Who could ask for more? 

It might almost seem that if Pompey reasoned this way, his old 
nose for victory had returned. By 53 B.C., Crassus' army was de- 
stroyed by the Parthians east of Syria and Crassus himself was 

But Caesar? No, Pompey 's luck had not returned. To the as- 
tonishment of everyone in Rome, Caesar, who, until then, had 
seemed to be nothing but a playboy and intriguer, turned out, in 
middle age (he was forty-four when he went to Gaul), to be a 
first-class military genius. He spent five years fighting the Gauls, 
annexing the vast territory they inhabited, conducting successful 
forays into Germany and Britain. He wrote up his adventures in 
his Commentaries for the Roman reading public, and suddenly 
Rome had a new military hero. — And Pompey, sitting in Italy, 
doing nothing, was nearly dead of frustration and envy. 

In 54 B.C., though, Julia died, and Pompey was no longer held 
back in his animus against Caesar. The senatorial aristocrats, now 
far more afraid of Caesar than of Pompey, flattered the latter, 
who promptly joined them and married a new wife, the daughter 
of one of the leading senators. 

When Caesar returned from Gaul in 50 B.C., the Senate ordered 
him to disband his armies and enter Italy alone. It was clear that 
if Caesar did so, he would be arrested and probably executed. 
What, then, if he defied the Senate and brought his army with 

"Fear not," said Pompey, confidently, "I have but to stamp my 
foot upon the ground and legions will rise up to support us." 

In 40 B.C, Caesar crossed the Rubicon River, which represented 
the boundary of Italy, and did so with his army. Pompey promptly 
stamped his foot — and nothing happened. Indeed, those soldiers 
stationed in Italy began to flock to Caesar's standards. Pompey 
and his senatorial allies were forced to flee, in humiliation, to 


Grimly, Caesar and his army followed them. 

In Greece, Pompey managed to collect a sizable army. Caesar, 
on the other hand, could only bring so many men across the sea 
and so Pompey now had the edge. He might have taken advantage 
of his superior numbers to cut Caesar off from his base and then 
stalk him carefully, without risking battle, and slowly wear him 
down and starve him out. 

Against this was the fact that the humiliated Pompey, still 
dreaming of the old days, was dying to defeat Caesar in open 
battle and show him the worth of a real general. Worse yet, the 
senatorial party insisted on a battle. So Pompey let himself be 
talked into one; after all, he outnumbered Caesar two to one. 

The battle was fought at Pharsalus in Thessaly on June 29, 
48 B.C. 

Pompey was counting on his cavalry in particular, a cavalry con- 
sisting of gallant young Roman aristocrats. Sure enough, at the 
start of the battle, Pompey's cavalry charged round the flank of 
Caesar's army and might well have wreaked havoc from the rear 
and cost Caesar the battle. Caesar, however, had foreseen this 
and had placed some picked men to meet the cavalry, with in- 
structions not to throw their lances but to use them to poke di- 
rectly at the faces of the horsemen. He felt that the aristocrats 
would not stand up to the danger of being disfigured and he was 
right. The cavalry broke. 

With Pompey's cavalry out, Caesar's hardened infantry broke 
through the more numerous but much softer Pompeian line and 
Pompey, unused to handling armies in trouble, fled. In one blow, 
his entire military reputation was destroyed and it was quite clear 
that it was Caesar, not Pompey, who was the real general. 

Pompey fled to the one Mediterranean land that was not yet 
entirely under Roman control — Egypt. But Egypt was in the 
midst of a civil war at the time. The boy-king, thirteen-year-old 
Ptolemy XII, was fighting against his older sister, Cleopatra, and 
the approach of Pompey created a problem. The politicians sup- 
porting young Ptolemy dared not turn Pompey away and earn 
the undying enmity of a Roman general who might yet win out. 



On the other hand, they dared not give him refuge and risk having 
Caesar support Cleopatra in revenge. 

So they let Pompey land — and assassinated him. 

And that was the end of Pompey, at the age of fifty-six. 

Up to the age of forty-two he had been uniformly successful; 
nothing he tried to do failed. After the age of forty-two he had 
been uniformly unsuccessful; nothing he tried to do succeeded. 

What happened at the age of forty-two? What circumstance 
took place in the interval represented earlier in the article by the 
line of asterisks that might "explain" this. Well, let's go back and 
fill in that line of asterisks. 

We are back in 64 B.C. 

Pompey is in Jerusalem, curious about the queer religion of the 
Jews. What odd things do they do besides celebrate a Sabbath? 
He began collecting information. 

There was the Temple, for instance. It was rather small and 
unimpressive by Roman standards but was venerated without 
limit by the Jews and differed from all other temples in the world 
by having no statue of a god or goddess inside. It seemed the Jews 
worshiped an invisible god. 

"Really?" said the amused Pompey. 

Actually, he was told, there was an innermost chamber in the 
Temple, the Holy of Holies, behind a veil. No one could ever go 
beyond the veil but the high priest, and he could only do so on 
the Day of Atonement. Some people said that the Jews secretly 
worshiped an ass's head there, but of course, the Jews themselves 
maintained that only the invisible presence of God was in that 

Pompey, unimpressed by superstition, decided there was only 
one way of finding out. He would look inside this secret chamber. 

The high priest was shocked, the Jews broke into agonized cries 
of dismay, but Pompey was adamant. He was curious and he had 
his army all around him. Who could stop him? So he entered the 
Holy of Holies. 



The Jews were undoubtedly certain that he would be struck by 
lightning or otherwise destroyed by an offended God, but he 

He came out again in perfect health. He had found nothing, 
apparently, and nothing had happened to him, apparently. * 

* In case you think I'm turning mystical myself, please reread the intro- 
duction to this chapter. 


I am, as it happens, doing a book on Lord Byron's narrative poem 
Don Juan.* The poem is an uninhibited satire in which Byron 
takes the opportunity to lash out at everything and everyone that 
displeased him. He is cruel to the point of sadism toward Britain's 
monarchs, toward its poet laureate, toward its greatest general, 
and so on. 

But those for whom he reserves his most savage sallies are his 
critics. Byron did not take to criticism kindly and he invariably 
struck back. 

Now, as far as I know, there is no such thing as a writer who 
takes to criticism kindly. Most of us, however, affect stoic un- 
concern and bleed in private. 

For myself, alas, stoic unconcern is impossible. My frank and 
ingenuous countenance is a blank page on which my every emo- 
tion is clearly written (I am told) and I don't think I have ever 
succeeded in playing the stoic for even half a second. Indeed, 
when I am criticized unfairly, everyone within earshot knows that 
I have been — and for as much as hours at a time. 

Naturally, when I recently published a two-volume book en- 
titled Asimov's Guide to Shakespeare, I tried to steel myself for 
inevitable events. It was bound to get into the hand of an occa- 
sional Shakespearian scholar who would come all over faint at the 
thought of someone outside their field daring to invade the sacred 

In fact, the very first review I received began: "What is Isaac 
Asimov, spinner of outer-space tales, doing — " 

Naturally, I read no further. The fact that I am a spinner of 
outer-space tales is utterly irrelevant to this particular book and 

* Because I want to, and because my publishers humor me. 



can only be mentioned because the reviewer thinks there is some- 
thing vaguely (or not so vaguely) beneath literary dignity in being 
a science fiction writer. 

I have sought a printable response for that and failed, so I'll 
pass on. 

A second review was much more interesting. It appeared in a 
Kentucky paper and was written by someone I will call Mr. X. 
It begins this way: "Isaac Asimov is associate professor of bio- 
chemistry at Boston University, and a prolific writer in many 
fields. I have read several of his books on science with the great- 
est attention and respect." 

So far, so good. I am delighted. 

But then, a very little while later, he says: "In this book, how- 
ever, he has left the sunlit paths of natural science for the 
treacherous bogs of literature — " 

What he objects to, it seems, is that I have annotated the plays, 
explained all the historical, legendary, and mythological refer- 
ences. It is a book of footnotes, so to speak, and he resents it. He 
points out that he thinks of "the language, the poetry, as the chief 
glory of Shakespeare's works." 

Well, who doesn't? I'm delighted that Mr. X is clever enough 
to understand the language and the poetry without any explana- 
tion from me. And if he doesn't need it, why should anyone else, 

Notice, though, that he doesn't scorn to follow me along "the 
sunlit paths of natural science." Indeed, he reads my books on 
science "with the greatest attention and respect." 

I'm glad he does and I can only presume that he is grateful that 
I take the trouble to footnote science so that he can get a fugi- 
tive hint of its beauties. 

Suppose, instead, I were to say to Mr. X, "The logarithm of 
two is a transcendental number: and, indeed, the logarithm of 
any integer to any integral base is transcendental except where 
the integer is equal to the base or to a power of that base." 

Mr. X might then, with justification, say, "What is a transcen- 


dental number, a logarithm, and, in this case, a power and a 

In fact, if he were a really deep thinker, he might ask, "What 
is two?" 

But suppose I answered then that my statement bore within it 
all the poetry and symmetry and beauty of mathematics ("Euclid 
alone has looked on beauty bare") and that to try to explain it 
would simply hack it up. And if Mr. X found trouble in under- 
standing it, too bad for him. He just wasn't as bright as I was, and 
he could go to blazes. 

But I don't answer that way. I explain such matters and many 
more, and go to a lot of trouble to do so, and then he reads those 
explanations with "the greatest attention and respect." 

Scientists generally recognize the importance of explaining 
science to the non-scientist. It is interesting, then, in a rather sad 
way, that there exist humanists who feel themselves to be 
proprietors of their field, who hug literature to themselves, who 
mumble "the language, the poetry," and who see no reason why 
it should be explained to anyone as long as they themselves can 
continue to sniff the ambrosia. 

Let us take a specific case. In the last act of The Merchant of 
Venice, Lorenzo and Jessica are enjoying an idyllic interlude at 
Portia's estate in Belmont, and Lorenzo says: 

Sit, Jessica. Look how the floor of heaven 
Is thick inlaid with patens of bright gold. 

There's not the smallest orb which thou behold'st 
But in his motion like an angel sings. 

Still quiring to the young-eyed cherubins; 

Such harmony is in immortal souls, 

But whilst this muddy vesture of decay 
Doth grossly close it in, we cannot hear it. 

I think this passage is beautiful, for I have as keen a sense of 
the beauty and poetry of words as Mr. X; perhaps (is it possible?) 
even keener. 



But what happens if someone says: "What are patens?" 

After all, it is not a very common word. Does it ruin the 
beauty of the passage to whisper in an aside, "Small disks"? 

Or what if someone asks, "What does he mean about these orbs 
singing like angels in their motion? What orbs? What motion? 
What singing?" 

Am I to understand that the proper answer is, "No! Just listen 
to the words and the beautiful flow of language, and don't ask 
such philistine questions." 

Does it spoil the beauty of Shakespeare's language to under- 
stand what he is saying? Or can it be that there are humanists 
who, qualified though they may be in esthetics, know little of the 
history of science, and don't know what Bill is saying and would 
rather not be asked. 

All right, then, let's use this as a test case. I am going to explain 
this passage in far greater detail than I did in my book, just to 
show how much there is to consider in these beautiful syllables- 

Anyone looking at the sky in a completely unsophisticated man- 
ner, without benefit of any astronomical training whatever, and 
willing to judge by appearances alone, is very likely to conclude 
that the Earth is covered by a smooth and flattened dome of some 
strong and solid material that is blue by day and black by night. 

Under that solid dome is the air and the floating clouds. Above 
it, he may decide, is another world of gods and angels where the 
immortal souls of men will rise after the body dies and decays. 

As a matter of fact, this is precisely the view of the early men 
of the Near East, for instance. On the second day of creation, 
says the Bible: "God said. Let there be a firmament in the midst 
of the waters, and let it divide the waters from the waters. And 
God made the firmament, and divided the waters which were 
under the firmament from the waters which were above the 
firmament" (Genesis 1:6-7). 

The word "firmament" is from the Latin word firmamentum, 
which means something solid and strong. This is a translation of 
the Greek word stereoma, which means something solid and 


strong, and that is a translation of the original Hebrew word raqia, 
which refers to a thin metallic bowl. 

In the biblical view there was water below the firmament 
(obviously) and water above it, too, to account for the rain. That 
is why in the time of Noah's flood, it is recorded that ". . . the 
fountains of the great deep [were] broken up, and the windows 
of heaven were opened" (Genesis 7:11). The expression might 
be accepted as metaphor, of course, but I'm sure that the unso- 
phisticated accepted it literally. 

But there’s no use laughing from the height of our own pain- 
fully gained hindsight. About 700 B.C., when the material of 
Genesis was first being collected, the thought that the sky was a 
solid vault with another world above it was a reasonable conclu- 
sion to come to from the evidence available. 

What's more, it would seem reasonable about 700 B.C. to sup- 
pose that the firmament stretched over but a limited portion of a 
flat Earth. One could see it come down and join the Earth tightly 
at the horizon. Few people in ancient times ever traveled far from 
home and the world to them was but a few miles in eveiy direc- 
tion. Even soldiers and merchants, who tramped longer distances, 
might feel the Earth was larger than it looked but that the world 
to the enlarged horizon was still flat, and still enclosed on all 
sides by the junction of firmament and ground. (This was also 
veiy much the medieval view and probably that of many unso- 
phisticated moderns.) 

The Greek philosophers, however, had come to the conclusion, 
for a number of valid reasons, that the Earth was not a more or 
less flat object of rather limited size, but a spherical object of suf- 
ficient size to dwarf the known world to small dimensions. 

The firmament, then, must stretch all around the globular 
Earth, and to do so symmetrically, it must be another, but much 
larger, sphere. The apparent flattening of the firmament over- 
head had to be an illusion (it is!) and the Greeks spoke of what 
we would call the "heavenly sphere" as opposed to the "terrestrial 

None of this, however, altered the concept of the firmament 



(or heavenly sphere) as made up of something hard and firm. 
What, then, were the stars? 

Naturally, the first thought was that the stars were exactly what 
they appeared to be: tiny, glowing disks embedded in the 
material of the firmament ("Look how the floor of heaven is thick 
inlaid with patens of bright gold"). 

The evidence in favor of this was that the stars did not fall 
down, as they would surely do if they were not firmly fixed to the 
heavenly sphere. Secondly, the stars moved about the Barth once 
eveiy twenty-four hours, with the North Star' as one pivot (the 
other being invisible behind the southern horizon), and did so 
all in one piece without altering their relative positions from night 
to night and from year to year. 

If the stars were suspended freely somewhere between the 
heavenly sphere and the Earth, and for some reason did not fall, 
surely they would either not move at all or, if they did, would 
move independently. No, it made much more sense to suppose 
them all fixed to the heavenly sphere, and to suppose that it was 
the heavenly sphere that turned, carrying all the stars with itself. 

But alas, this interpretation of the heavens — beautiful and 
austerely simple — did not account for everything. 

As it happened, the Moon was clearly not imbedded in the 
heavenly sphere, for it did not maintain a fixed position relative 
to the star's. It was at a particular' distance from a particular' star 
one night, farther east the next night, still farther east the one 
afterward. It moved steadily west to east in such a way as to make 
a complete circuit of the starry sky in a little over twenty-seven 

The Sun moved from west to east, too, relative to the star's, 
though much more slowly. Its motion couldn't be watched di- 
rectly, of course, since no star's were visible in its neighborhood 
by which its position might be fixed. However, the nighttime con- 
figuration of star's shifted from night to night because, clearly, the 
Sun moved and blotted out slightly different portions of the sky 
from day to day. In that manner it could be determined that the 
Sun seemed to make a circuit of the sky in a little over 365 days. 



If the Sun and the Moon were the only bodies to be exceptional, 
this might not be too bad. After all, they were veiy much differ- 
ent from the stars and could not be expected to follow the same 

Thus the Hebrews, in their creation myth, treated the Sun 
and Moon as special cases. On the fourth day of creation, "God 
made two great lights; the greater light to rule the day, and the 
lesser light to rale the night: he made the stars also" (Genesis 

1 : 16 ). 

It see ms amusing to us today to have the stars dismissed in so 
offhanded a fashion, but it makes perfect sense in the light of the 
Hebrew knowledge of the day. The stars were all imbedded in 
the firmament and they served only as a background against which 
the motions of the Sun and the Moon could be studied. 

But then it turned out that certain of the brighter stars were 
also anomalous in their motions and shifted positions against the 
background of the other stars. In fact, their motion was even 
stranger than that of the Moon and the Sun, for, though they 
moved west to east most of the time, relative to the stars, as the 
Moon and the Sun did, they occasionally would turn about and 
move east to west. Very puzzling! 

The Greeks called these stars planetes, meaning "wanderers," 
as compared with the "fixed stars." The Greek word has become 
"planet" to us and seven of them were recognized. These included 
the five bright stars which we now call Mercury, Venus, Mars, 
Jupiter, and Saturn, and, of course, the Sun and the Moon. 

What to do with them? Well, like the stars, the planets did not 
fall and like the stars they moved about the Earth. Therefore, 
li ke the stars, they had to be embedded in a sphere. Since each 
of the seven planets moved at a different speed and in a differ- 
ent fashion, each had to have a separate sphere, one nested inside 
the other, and all nested inside the sphere of the stars. 

Thus there arose the notion not of the heavenly sphere, but of 
the heavenly spheres, plural. 

But there was only one heavenly sphere that could be seen — 
the blue sphere of the firmament. The fact that the other spheres 



were invisible was no argument, however, for their non-existence, 
merely for their transparency. They were sometimes called "the 
crystalline spheres," where the word "crystalline" was used in its 
older meaning as "transparent." 

The Greeks then set about trying to calculate where the differ- 
ent spheres were pivoted and how they must turn in order to 
cause each planet to move in the precise fashion in which it was 
observed to move. Endless complications had to be added in order 
to match theory with observation, but for two thousand years the 
complicated theory of the crystalline spheres held good, not be- 
cause men of thought were perversely stupid, but because nothing 
else so well fit the appearances. 

Even when Copernicus suggested that the Sun, not the Earth, 
was the center of the universe, he didn't abolish the spheres. He 
merely had them surrounding the Sun, with the Earth itself em- 
bedded in one of them. It was only with Johannes Kepler — 

But never mind that. The details of the motions of the crys- 
talline spheres don't concern us in this article. Let us instead con- 
sider an apparently simpler question: In what order are the 
spheres nested? If we were to travel outward from Earth, which 
sphere would we come to first, which next, and so on. 

The Greeks made the logical deduction that the closest sphere 
would be smallest and would therefore make a complete turn in 
the briefest time. Since the Moon made a complete circle against 
the stars in about four weeks (a far shorter time than any other 
planet managed to run the course), its sphere must be closest. 

Arguing in this manner, the Greeks decided the next closest 
sphere was that of Mercury; then, in order, Venus, the Sun, Mars, 
Jupiter, and Saturn. And finally, of course, there was the sphere 
of the stars. 

And how far apart were the spheres and what were their actual 
distances from the Earth? 

That, unfortunately, was beyond the Greeks. To be sure, the 
Greek astronomer Hipparchus, about 150 B.C, used a perfectly 
valid method (after the still earlier astronomer Aristarchus) for 
determining the distance of the Moon, and had placed it at a 




distance of thirty times the Barth’s diameter, which is correct, 
but the distance of no other heavenly body was determined with 
reasonable accuracy until the seventeenth century. 

Now the scene switches. About 520 B.C, the Greek philosopher 
Pythagoras was plucking strings, and found that he could evoke 
notes that harmonized well together if he used strings whose 
lengths were simply related. One string might be twice the length 
of another; or three strings might have lengths that were in the 
ratio of 3:4:5. 

The details are irrelevant, but to Pythagoras it seemed highly 
significant that there should be a connection between pleasing 
sounds and small whole numbers. It fit in with his rather mystical 
notion that everything in the universe was related to simple ratios 
and numbers. 

Those who followed in his footsteps after his death accentuated 
the mysticism and it seemed to the Pythagoreans that they now 
had a way of deciding not only the how of planets, but the why 
as well. Since numbers governed the universe, one ought to be 
able to deduce the way in which the universe ought to be con- 

For instance, 10 was a particularly impressive number. (Why? 
Well, for one thing, 1 + 2 + 3 + 4= 10, and this see ms to have some 
mystical value.) In order, then, for the universe to function well, 
it had to be composed of ten spheres. 

Of course, there were only eight spheres, one for the stars and 
one for each of the seven planets, but that didn't stop the 
Pythagoreans. They decided that the Earth moved around some 
central fire of which the Sun was only a reflection, and worked 
up a reason for explaining why the central fire was invisible. That 
added a ninth sphere for the Barth. In addition, they imagined 
another planet on the opposite side of the central fire, a "counter- 
Earth". The counter-Earth kept pace with the Barth and stayed 
always beyond the central fire and was thus never seen. Its sphere 
was the tenth. 

In addition, the Pythagoreans thought that the spheres were 



nested inside each other in such a way that their distances of 
separation bore simple ratios to one another and produced har- 
monious notes in their motion as a result (like the plucking of 
strings of simply related lengths). Originally, I imagine, the 
Pythagoreans may have advanced this notion of harmonious notes 
only as a metaphor to represent the simply related distances, but 
later mystics accepted the notes as literally existent. They became 
"the music of the spheres." 

Of course, no one ever heard any music from the sky, so it had 
to be assumed to be inaudible to men on Earth. It is this notion 
that causes Shakespeare to speak of an orb that "in his motion 
like an angel sings" but with sounds that can be heard only in 
heaven ("Such harmony is in immortal souls"). While men's souls 
are still draped in their earthly bodies, they are deaf to it ("whilst 
this muddy vesture of decay doth grossly close it in, we cannot 
hear it"). 

Well, then, does understanding Lorenzo's speech in terms of 
ancient astronomy spoil its beauty? Does it not seem that to 
understand him adds to the interest? Does it not remove the nag- 
ging question of "But what does it mean?" that otherwise dis- 
tracts from an appreciation of the passage? 

It may be, of course, that Mr. X is the kind who never asks 
"But what does it mean?" It may be that for him understanding 
is irrelevant. If so, he and I are not soul mates. It may even be 
that Mr. X is the kind of obscurantist who finds that understand- 
ing decreases beauty. If so, he and I are even less soul mates. 

And yet, let me point out that there is something in this very 
passage that could be of interest to Shakespearian scholars if 
they thoroughly understood what Shakespeare was talking about. 

As almost everyone knows, there are many who feel that Shake- 
speare did not write the plays attributed to him. They feel that 
someone else did, with the person most frequently credited being 
Francis Bacon, who was an almost exact contemporary of Shake- 

The argument very often heard is that Shakespeare was just a 



fellow from the provinces with veiy little education and that he 
could not possibly have written so profoundly learned a set of 
plays. Bacon, on the other hand, was a great philosopher and one 
of the most intensely educated people of his time. Bacon, there- 
fore, could easily have written the plays. 

Shakespearian scholars, when they argue the matter at all, are 
forced to maintain that Shakespeare was much better educated 
than he is given credit for being and that therefore he was learned 
enough to write his plays. Since virtually nothing is known of 
Shakespeare’s life, the argument will never be settled in that 

Why not turn matters around, then, and argue that Bacon was 
too educated to write Shakespeare's plays, that there exist error's 
in the plays that Bacon could never possibly have made and that 
would just suit an insufficiently educated fellow from the sticks? 

Consider Lorenzo's speech. Lorenzo is talking about the stars; 
these are the "patens of bright gold" with which "the floor of 
heaven is thick inlaid." Lorenzo (hence Shakespeare) seems to 
think that each star' has a separate sphere and that each gives out 
its own note ("There's not the smallest orb which thou behold'st / 
But in his motion like an angel sings"). 

Lest you think I'm misinterpreting the speech, let's take a 
clearer case. 

In Act II of A Midsummer Night's Dream, Oberon is remind- 
ing Puck of a time they listened to a mermaid who sang with such 
supernal beauty that 

. . . the rude sea grew civil at her song. 

And certain stars shot madly from their spheres. 

To hear the sea maid's music. 

The use of the plural "spheres" shows again that Shakespeare 
thinks that each star' has its separate sphere. 

This is wrong. There is a sphere for each planet; one for the 
Earth itself, if you like; one for the counter-Earth; one for any 
imaginary planet you wish. However, all the ancient theories 



agreed that the "fixed stars" were all embedded in a single sphere. 

To imagine separate spheres for each star, as Shakespeare does 
more than once in his plays, is to display a lack of knowledge of 
Greek astronomy. This is a lack of knowledge that Francis Bacon 
could not possibly have displayed; hence we might fairly argue 
that Francis Bacon could not possibly have written Shakespeare's 

Well, don't get me wrong. I don't want to imply that I received 
only bad reviews for my Guide to Shakespeare. Actually, most 
of the reviews were quite complimentary and were an entire 
pleasure to read. 

Just the same, I had better start preparing myself for the oc- 
casional review by the "outraged specialist" type that I will surely 
get when Asimov's Annotated "Don Juan" is published. 

F — The Problem of Population 

16— STOP! 

As some of my Gentle Readers may know, 1 am an after-dinner 
speaker when I can be persuaded to be one. (For the information 
of prospective persuaders, I may as well state at once that the best 
persuasion is a large check.) 

As a speaker, 1 must be introduced, of course, and introductions 
vary in quality. It's not difficult to see that a short introduction 
is better than a long one, since much preliminary talk dulls the 
edge of the audience and makes the speaker's task harder. 

Again, a dull introduction is better than a witty one, since a 
speaker can easily suffer by contrast with preliminary wit, and an 
audience which might otherwise be receptive enough becomes 
critical after the joy of the introduction. 

Needless to say, then, the veiy worst possible introduction a 
speaker can have is one that is both long and witty, and on the 
night of April 20, 1970, at Pennsylvania State University, that is 
exactly what I got. 

Phil Klass (far better known to science fiction fans as William 
Tenn) is associate professor of English at Penn State and it 
naturally fell to him to introduce me. With an evil smile on his 
face, he got up and delivered an impassioned address that went 
on for fifteen minutes and that had the audience of some twelve 
hundred people rocking with laughter (at my expense, naturally). 
As he went on, a kind of grimness settled about my soul. 1 
couldn't possibly follow him; he was too good. Naturally, 1 de- 
cided to kill him as soon as I got my hands on him, but fust 1 had 
to live through my own talk. 

And then at the veiy last minute, Phil (I'm sure, unintention- 
ally) saved me. He concluded his talk by saying, "But don't let 


me give you the idea that Asimov is a Renaissance Man. He has 
never, after all, sung Rigoletto at the Metropolitan Opera." 

1 brightened up at once, rose smiling from my seat, and 
mounted the stage. 1 waited for the polite opening applause to 
die down and, without preliminary, launched my resonant voice 
into "Bella figlia delTamore — " the opening of the famous Quartet 
from Rigoletto. 

It was the first time 1 ever got the biggest laugh of the entire 
evening with my first four words, and after that I had no trouble 
at all. 

1 tell you all this because in April 1970,1 gave nine ta lk s which, 
despite Rigoletto, were not funny at all. It was the month in 
which the first Earth Day was celebrated, and eveiy one of my 
talks dealt, in whole or in part, with the coming catastrophe. 

1 have discussed that catastrophe in the final chapters of a 
previous volume. The Stars in Their Courses (Doubleday, 1971), 
and I have made it quite plain that in my opinion the first order 
of business is a halt to the population increase on Earth. Without 
such a halt right away, none of mankind's problems can be 
solved under any conditions: none! 

The question then is: How can the population increase be 

Since this is now the prime question and, indeed, the only 
relevant question that futurists have to face, and since science 
fiction writers were futurists long before the word was invented, 
and since 1 am self-admittedly one of the leading science fiction 
writers, I consider it my duty to tiy to answer this question. 

To begin with, let us admit there are only two general ways of 
bringing about a halt in the population increase: we might in- 
crease the death rate, or we might decrease the birth rate. (We 
might, conceivably, do both, but the two are independent and 
can be discussed separately.) 

Let's stall with the increase of the death rate first and consider 
all the variations on the theme: 



A - Increase in the death rate 
1 -Natural increase 

This is the system that has been in use for all species since 
life began. It is the system that served to limit human population 
throughout its history. When food grew scarce, human beings 
starved to death, were easier prey for disease in their famished 
condition, fought each other and killed in order to gain access to 
what food supplies there were, led armies into other regions where 
food was more plentiful. For all these reasons the death rate rose 
precipitously and population fell to match the food supply. 

We have here the "four horsemen of the Apocalypse" (see the 
sixth chapter of the biblical Book of Revelation) — war, civil strife, 
famine, and pestilence. 

Modem science has greatly weakened the force of the third and 
fourth horsemen, and both famine and pestilence are not what 
they once were. This in itself has amazingly lowered the death rate 
horn what it was in all the millennia before 1850 and is the major 
reason for the explosiveness with which population has increased 

We can well imagine, however, that if the population continues 
to soar' for another generation, the efforts of science will crack 
under the strain. All four horsemen will regain their ascendancy; 
the death rate will zoom upward. 

Possibly one might be objective about this and say: Well, this 
is the way the game of life is played. The fittest will survive and 
mankind will continue stronger than ever, for the winnowing-out 
it has received. 

Not at all! There might have been some validity to this view, 
for all its inhumanity, if mankind were armed with stone axes 
and spears, or even with machine guns and tanks. Unfortunately, 
we have nuclear weapons at our disposal and when the four horse- 
men stall out on their horrid ride, the H-bombs will surely be 

Mankind, living in the tattered remnants of a world tom by 
thermonuclear war, will not be stronger than ever. It will be living 



not only in the ruins of a destroyed technology, but in the midst 
of a dangerously poisoned soil, sea, and atmosphere which may 
no longer be able to support vertebrate life at all. 

We'll need something better. 

2 - Directed general increase 
a - Involuntary 

Instead of waiting for the course of events to enforce a cata- 
strophic increase in death rate, we might blow off steam by 
randomly killing off part of the population from year to year. 
Suppose that preliminary estimates during a census year make it 
seem that the world population is 10 per cent above optimum. 
In that case, take the census and shoot every tenth person 

About the only thing that can be said about this method is that 
it is perhaps a little better than a thermonuclear war. I don't think 
any sane man would consider it if any other alternative existed 
at all. 

b - Voluntary 

Random killing might be made voluntary if one constructed a 
suicide-centered society.** In such a society, suicide must be made 
to seem attractive, either through the effective promise of an 
afterlife or through the more material offer of financial benefits 
to the family left behind. 

Somehow, though, I doubt that under any persuasion not in- 
volving physical constraint or emotional inhumanity, enough peo- 
ple will kill themselves to halt the population increase. Even if 
enough did, the kind of society that would place the accent on 
death with sufficient firmness to bring it about would undoubtedly 
be too unbearably morbid for the health of the species. 

* "The Census Takers," an excellent science fiction story by Frederik 
Pohl, actually uses this situation. 

** Gore Vidal's Messiah had something of this sort. 



3 -Directed special increase 
a - Inferiority 

But if we must kill, would it be possible to neutralize some of 
the horror by making murder serve some useful purpose. Suppose 
we kill off or (more humanely) sterilize that portion of the 
population that contributes least to mankind, the "inferior" por- 
tion, in other words. 

Indeed, such a policy has been put into practice on numerous 
occasions, though not usually out of a set, reasoned-out popula- 
tion strategy. Throughout Earth's history, a conquering nation 
has usually made the calm assumption that its own people were 
superior to the conquered people, who were therefore killed or 
enslaved as a matter of course. Under conditions of famine the 
conquered peasantry would surely die in greater proportion than 
the conquering aristocracy. 

Conquerors varied in inhumanity. In ancient times, the Assyr- 
ians were most noted for the callous manner in which they would 
destroy the entire male population of captured cities; and in 
medieval times, the Mongols made a name for themselves in the 
same fashion. In modern times, the Germans under Hitler, more 
consciously and deliberately, set about destroying those whom they 
considered members of inferior races. 

This policy can never be popular except with those who have 
the power and the inhumanity to declare themselves superior 
(and not usually with all of those either). The majority of man- 
kind is bound to be among the conquered and the inferior and 
their approval is not to be expected. The Assyrians, Mongols, 
and Nazis were all greeted with nearly universal execration both 
in their own times and thereafter. 

There are individuals whom the world generally would con- 
sider inferior — the congenital idiot, the psychopathic murderer, 
and so on — but the numbers of such people are too few to matter. 

b-Old age 

Perhaps then people can be killed off according to some 
category that isn't as subjective as superiority-inferiority. What 



about the very old? They still eat; they are still drains on the 
culture; yet they give back very little. 

There have been cultures which killed those aged members that 
could not carry their own weight (the Eskimos, for instance). 
Before late modern times, however, there was usually little pres- 
sure in this direction, since very few members of a society man- 
aged to live long enough to be too old to be worth their keep. 
Indeed, the very few aged members might even be valuable as the 
repositories of tradition and custom. 

Not so nowadays. With the rise in life expectancy to seventy, 
the "senior citizen" is far more numerous in absolute numbers 
and in proportion than ever before. Ought all those who reach 
sixty-five, say, be painlessly killed? If this applies to all humans 
without exception there would be no subjective choice and no 
question of superiority-inferiority. 

But what good would it do? The men and women thus killed 
are past the child-bearing age and have already done their damage. 
Such euthanasia will make the population younger but not do 
one thing to stop the population increase. 

c- Infants 

Then why not the other end of the age scale? Why not kill 
babies? Infanticide has been a common enough method of popu- 
lation control in primitive societies, and in some not so primitive. 
Usually, it is the girl babies that are allowed to die, and, to be sure, 
that is as it should be. 

I hasten to say that I do not make the last statement out of 
anti-female animus. It is just that it is the female who is the bottle- 
neck. Compare the female, producing thirteen eggs a year and 
fertile for limited periods each month, with the male, producing 
millions of sperm each day and nearly continuously on tap. A 
hundred thousand women will produce the same number of babies 
a year whether there are ten thousand men at their free disposal 
or a million men. 

Actually, there are some points in favor of infanticide. For one 



thing, it definitely works. Carried out with inhuman efficiency, 
it could put an end to the human race altogether in the space of 
a century. It can be argued moreover that a newborn baby is only 
minimally conscious and doesn't suffer the agonies of apprehen- 
sion; that he as yet lacks personality and that no emotional ties 
have had a chance to form about him. 

And yet, infanticide isn't pleasant. Babies are helpless and ap- 
pealing and a society that can bring itself to slaughter them is 
perhaps too callous and inhumane to serve mankind generally. 
Besides, we cannot kill all babies, only some of them, and at once 
an element of choice enters. Which babies? The Spartans killed all 
those that didn't meet their standards of physical fitness and in 
general the matter of superiority-inferiority enters with all its 

d- Fetuses 

What about pre -birth infanticide — in short, abortion. Fetuses 
are not independently living and society's conscience might be 
quieted by maintaining they are therefore not truly alive. They 
are not killed, they are merely "aborted," prevented from gaining 
full life. 

Of all forms of raising the death rate, abortion would seem the 
least inhumane, the least abhorrent. At the present moment, in 
fact, there are movements all over the world, and not least in the 
United States to legalize abortion. 

And yet if one argues that killing a baby is not quite as bad as 
killing a grown man, and killing a fetus not quite as bad as killing 
a baby, why not go one step farther, and kill the fetus at the very 
earliest moment? Why not kill it before it has become a fetus, 
before conception has taken place? 

It seems to me then that any humane person, considering all 
the various methods of raising the death rate must end by deciding 
that the best method is to prevent conception; that is, to lower 
the birth rate. Let's consider that next. 

If we consider the different ways of decreasing the birth rate. 



we can see that, to begin with, they fall in two broad groups: 
voluntary and involuntary. 

B -Decrease in birth rate 
1 - Voluntary 

Ideally, this is the situation most acceptable to a humane 
person. If the population increase must be halted, let everyone 
agree to and voluntarily practice the limitation of children. 

Everyone might simply agree to have no more than two chil- 
dren. It would be one, then two, then STOP! 

If this came to pass, not only would the population increase 
come to a halt* it would begin to decrease. After all, not all 
couples would have two children. Some, through choice or circum- 
stance, would have only one child and some even none at all. 
Furthermore, of the babies that were born, some would be bound 
to die before having a chance to become adults and have babies of 
their own. 

With each generation under the two-baby system, then, the 
total population of mankind would decrease substantially. 

I do not consider this a bad thing at all, for I feel that the Earth 
is already, at this moment, seriously overpopulated. I could argue, 
and have, that a closer approach to the ideal population of Earth 
would be one billion people, and this goal would allow several 
generations of shrinkage. In a rational society, without war or 
threat of war, it seems to me that a billion people could be sup- 
ported indefinitely. 

If the population threatened to drop below a billion, it would 
be the easiest thing in the world to raise the permitted number 
of babies to three per couple. Enough couples would undoubtedly 
take advantage of permission to have a third child to raise the 
population quickly. 

I would anticipate that under a humane world government, 

* Provided the life expectancy doesn't increase drastically. If it did. there 
would be a continued accumulation of old people. It might be just as well 
not to labor to increase that expectancy above the level that now exists. It 
embarrasses me to say so but I see no way out. 



a decennial census applied to the whole world would, on each 
occasion, serve to guide the decision whether, for the next ten 
years, third children would be asked for or not. 

Such a system would work marvelously well, if it were adopted, 
but would it be? Would individuals limit births voluntarily? I am 
cynical enough to think not. 

In the first place, where two is the desired number of babies 
per couple, it is so much easier to far overshoot the mark than 
far undershoot it. A particular couple can, without biological dif- 
ficulty, have a dozen children, ten above par. No couple, however, 
no matter how conscientious, can have fewer than zero children, 
or two under par. 

This means that for every socially unfeeling couple with a dozen 
children, five couples must deprive themselves of children alto- 
gether to redress the balance. 

Furthermore, I suspect that those families who, on a strictly 
voluntary basis, choose to have many children, are apt to be 
drawn from those with less social consciousness, less feeling of 
responsibility — for whatever reason. Each generation will con- 
tribute to the next generation in a most unbalanced fashion. 

This would, in fact, very likely cause an utter breakdown in the 
voluntary system in short order, for there will be resentment and 
fear on the part of the socially conscious. The socially conscious 
will easily convince themselves that it is precisely the ignorant, 
the inferior, the undeserving who are breeding and they may feel 
that it is important for them to supply the world with their own, 
much-more-desirable offspring. 

It is even rather likely that, as long as birth control is purely 
voluntary, it will be negated out of local sub-planetary considera- 

In Canada, for instance, the birth rate is higher among the 
French-speaking portion of the population than among the 
English-speaking portion. I am sure that there are those on both 
sides of the fence who calculate, with hope or with fear, that the 
French-Canadians will eventually dominate the land out of sheer 
natural increase. 


The French-Canadians might be loath to adopt voluntary birth 
control and lose the chance of domination, while the English- 
Canadians might be loath to adopt it and perhaps hand over 
the domination all the more quickly to a still breeding French- 
Canadian population. 

The situation might be similar within the United States, where 
Blacks have a higher birth rate than whites; or in Israel, where 
the Arabs have a higher birth rate than the Jews; or in almost 
any country with a non-homogeneous population. 

It is not only inside a country where such questions would 
arise. The Greeks would not want to fall too far behind the 
Bulgarians in population; the Belgians too far behind the Dutch; 
the Indians too far behind the Chinese; and so on and so on. 

Each nation, each group within a nation, would watch its 
neighbors and would attempt to retain the upper hand for itself 
or (which is the same thing) prevent the neighbor from gaining 
the upper hand. And, in the name of patriotism, nationalism, 
racism, voluntary birth control would fail and mankind would be 

2 - Involuntary 

Ought we then not merely ask couples not to have more than 
two children; ought we to tell them? 

Suppose, for instance, that all babies were carefully registered 
and that every time a woman had a second baby, the first one 
being still alive, she be routinely sterilized before being released 
from the hospital. 

Why women? you might ask. Why not men, for whom the 
operation is simpler. 

My choice of women is not the result of male chauvinism on 
my part but only because women, as I said before, are the bottle- 
neck in reproduction. Sterilizing some males will do no good if 
the rest merely work harder at it, while sterilizing females must 
force the birth rate down. Then, too, one knows when a female 
has two children; one can only guess at it with males. Finally, it is 



the woman, not the man, who is on the hospital table at the time 
of birth. 

But would such involuntary birth control work? Or would it 
arouse such resentment that the world would constantly rock 
with insurrection, that women would have their babies in secret, 
that the government would be forced into more extremes of 
tyranny constantly. 

Somehow I suspect that the system would indeed break down 
if the process were not carried through without exception. 

There would be a strong temptation, I suppose, to work out 
some sort of regulations whereby some people would be allowed 
three children or even four, while others might be allowed only 
one or even none at all. You might argue that college graduates 
ought to have more children than morons should; proven achievers, 
more than idle dreamers; athletes, more than diabetics; and so on. 

Unfortunately, I don't think that any graduated system, how- 
ever impartially and sensibly carried through, can possibly succeed. 

Whatever the arrangement, there will be an outcry that group X 
is favored over group Y. At least group Y will say so and will 
gather information to prove that group X is in control of the 
World Population Council. Using the same statistics and informa- 
tion, group X will insist that group Y is being favored. 

The only possible solution, however wasteful, would be to al- 
low no exceptions at all for any reason. Let the "fit" have no more 
children than the "unfit" (no less, either), in whatever way your 
own emotions and prejudices happen to define "fit" and "unfit." 

Then, when the population is reduced to the proper level and 
the Earth has had several generations of experience with a humane 
world government, propositions for grading birth numbers and 
improving the quality of humanity without increasing its quantity 
may be entertained. 

Yet I must admit that the use of the knife, the inexorable push 
of governmental surgery is unpalatable to me and would probably 
be unpalatable to many people. If there were only some way to 
make voluntary compliance as surefire as the involuntariness of 
sterilization, I would prefer that. 



Could we leave people the choice; could we let them choose 
the additional child if they wish — but make it prohibitive for 
various reasons? Could we find pressures as inexorable as the 
knife, yet leaving the human body and, therefore, human dignity 

3 -Voluntary, with encouragement 

Let's go back to voluntary birth limitation, but now let's not 
make it entirely voluntary. Let's set up some stiff penalties for 
lack of co-operation. 

To begin with, reverse the philosophy of the income tax. At 
present births are encouraged by income tax deductions. Suppose 
there are penalties instead. Your tax would go up slightly with 
one child, up again slightly with two, and then up prohibitively 
with three. 

In other words, couples are bribed not to have children. 

There are other forms of bribes. When a third child is born, 
a husband might suffer a pay cut, or lose his job altogether and 
be forced to go on welfare. A three-child family may lose medical 
plan privileges, be barred from air flight, be ostracized by other 

This is all very cruel but in the world today that third child 
is a social felony. 

Is that kind of pressure better than the knife? Will it force man- 
kind less strongly into secret births, whole hidden colonies of 
forbidden children? Will the third children who are bom be mis- 
treated or killed? Will the rule discriminate in favor of the rich? 

I don't know, but I can't think of anything better. It seems to 
me that the need is overwhelming and the time is now. Let's be- 
gin at once to persuade people, one way or another, not to have 
babies, to begin building the social pressures against large families. 
It is that, or the death of civilization and of billions of human 
beings with it.* 

Only one thing — 

* In case your curiosity has grown unbearable, I myself have two children. 
I will have no more. 



Suppose we adopt this final alternative and suppose humanity 
generally and genuinely accepts it. People everywhere honestly 
intend to have no more than two children. Each couple which 
has its two children must now decide (without compulsory 
sterilization, mind you) to figure out a way not to have the third. 

How? What alternatives are open to them? — For remember, if 
there are no reasonable alternatives, we are back to compulsory 
sterilization. — Or doom. 



Sometimes I wish 1 were small enough to know when I've hap- 
pened to say something small so that I can get it down on paper 
and notarize it, as proof for posterity. 

For instance, back in 1952, I was listening to the news of the 
election-day Eisenhower landslide with considerable gloom* when 
a ray of sunshine penetrated the darkness. 

It seemed a young Democrat had just won his election to the 
Senate by a co mf ortable margin in the face of the tidal wave in 
the other direction at the presidential level. He was shown thank- 
ing his election workers and, in doing so, displayed such irresisti- 
ble charm that 1 turned to my wife and said: 

"If he weren't a Catholic, he'd be the next President of the 
United States, after Eisenhower." 

You're ahead of me, I know, but that young man was John F. 
Kennedy and I was remarkably prescient. Unfortunately, I have 
no record of the remark and my wife — the only witness — doesn't 
remember it. 

On the other hand, at about the same time, in the early 1950's, 
I said, in the course of a discussion at a social gathering, "This is 
the last generation in which the unrestricted right to breed will 
remain unquestioned. After this, birth control will be enforced." 

"What about the Roman Catholic Church?" someone asked me. 

"The Roman Catholic Church," I said, "will have no choice 
but to go along." 

I was hooted down by unanimous consensus and it was the 
general feeling that being a science fiction writer had gone to 
my head — but I still stand on what I said nearly twenty years ago. 

* I will hide nothing from you. I am a Democrat. 



So we'll limit births for reasons 1 explained in the previous 

— But how? 

There are many methods of birth control practiced. There is 
abstention and chastity, for example. (Don't laugh! For some 
people, this works, and we are in no position to turn down the 
help offered by any method, however minimal.) There is the 
rhythm method, of choosing, or hying to choose, that time of 
the month when a woman is not ovulating. There is the practice 
of withdrawal, or of surgical and permanent sterilization, or of 
chemical and temporary sterilization, or of mechanical intercep- 
tion, and so on. 

All have their value as far as birth control is concerned; all have 
their disadvantages; no one method will do the trick by voluntary 
acceptance; perhaps even all together will not do the trick. 

Nevertheless, we must try, and if anyone can think of some 
technique that is not being tried but ought to be, it is his duty, 
in this crisis facing mankind to advance it as forcefully as he can. 
This 1 intend to do. 

The real enemy, as 1 see it, is social pressure, which is the 
strongest human force in the world. Love laughs at locksmiths 
and may flourish under the severest legal condemnations, but it 
is love indeed that can persist under no punishment worse than 
the cold-hearted ostracism of society. 

Social pressure is irrepressible. The rebels who stand firmly 
against the Establishment and who object to all the moss-grown 
mores of yore, quickly develop a subculture with mores of its own 
which they do not, and dare not, violate. 

And it is social pressure, inexorable social pressure, that dictates 
that people shall have children — lots of children — the more chil- 
dren the better. 

There is reason for it. Despite what many think, the conven- 
tions of society are not invented merely to annoy and confuse, 
or out of a perverse delight in stupidity. They make sense — in the 
context of the times in which they originate. 

Until the nineteenth century, there was virtually no place on 


Earth and virtually no time in history in which life expectancy 
was greater than thirty-five years. In most places and most times 
it was considerably less. There was virtually no place and no time 
in which infant mortality wasn't tenifyingly high. It was not the 
death of children that was surprising, but their survival. 

Through all the ages of high infant mortality and low life 
expectancy, it stood to reason that each family had to have as 
many children as possible. This was not because each family sat 
down and worried about the future of mankind in the abstract. 
Not at all; it was because in a tribal society, the family is the social 
and cultural unit, and as many young as possible were necessary 
to cany on the work of herding or farming or whatever, while 
standing to their weapons to keep off other tribes at odd moments. 
And it took all the children the women could have to supply the 
necessary manpower. 

With death so prevalent through hunger, disease, and warfare, 
the problem of overpopulation did not arise. If, unexpectedly, a 
tribe's numbers did increase substantially, they could always move 
outward and fall on the next tribe. It was the withering and 
extinction of the tribe that seemed the greater danger. 

Consequently, social pressures were in favor of children, and 
naturally and rightly so. 

We needn't go off into anthropological byways to see evidence 
of this; we have it at our fingertips in the Bible — the most impor- 
tant single source of social pressure in Western civilization. (And 
this is crucial, for it is Western culture that controls the Earth 
militarily, and Western culture that will have to lead the way in 
population policy.) 

The first recorded statement of God to humanity after its 
creation is: "And God blessed them and said unto them. Be fruit- 
ful and multiply, and replenish the Earth — " (Genesis 1:28). 

On a number of occasions thereafter, the Bible records the fact 
that the inability to bear children is considered an enormous 
calamity. God promises Abram that he will be taken care of, say- 
ing, "... Fear not, Abram: I am thy shield and thy exceeding 
great reward" (Genesis 15:1). But Abram can find no comfort in 



this and says, . . Lord God, what wilt thou give me, seeing 1 
go childless . . (Genesis 15:2). 

In fact, childlessness was viewed as divine punishment. Thus, 
Jacob married two sisters: Leah and Rachel. He had wanted only 
Rachel but had been forced to take Leah through a trick. As a 
result, he showed considerable favoritism and of this God ap- 
parently disapproved; "And when the Lord saw that Leah was 
hated, he opened her womb: but Rachel was barren" (Genesis 

Naturally, Rachel was upset. "And when Rachel saw that she 
bare Jacob no children, Rachel envied her sister; and said unto 
Jacob, Give me children, or else I die" (Genesis 30:1). 

There is the case of Hannah, who was barren, despite constant 
prayer; and who was miserable over it, despite the faithful love 
of her husband, who overlooked her barrenness (which made her 
worthless in a tribal sense and which placed her under strong 
suspicion of sinfulness) and expressed his love for her most 
touchingly: "Then said Elkanah her husband to her, Hannah, 
why weepest thou? and why eatest thou not? and why is thy 
heart grieved? am not I better to thee than ten sons?" (1 Samuel 
1 : 8 ). 

But Hannah perseveres in prayer and conceives at last, bearing 
Samuel. The second chapter of the book contains her triumphant 
song of celebration. 

A particularly clear indication that barrenness is the punish- 
ment of sin arises in connection with the history of David. David 
had brought the Ar k of the Covenant into Jerusalem and, in 
celebration, had participated in the ritualistic, orgiastic dance of 
celebration, one in which (the Bible is not clear) there may have 
been strong fertility-rite components. David's wife, Michal, dis- 
approved strongly, saying sarcastically, "... How glorious was 
the king of Israel to day, who uncovered himself to day in the 
eyes of the handmaids of his servants, as one of the vain fellows 
shamelessly uncovereth hi ms elf I" (2 Samuel 6:20). 

This criticism displeased David and, apparently, God as well. 



for "Therefore Michal the daughter of Saul had no child unto 
the day of her death" (2 Samuel 6:23). 

So strong was the tribal push for children that if a wife were 
barren, she herself might take the initiative of forcing her hus- 
band to impregnate a servant of her own, that she might have 
the credit of children by surrogate. Thus, when Abram's wife, 
Sarai, proved barren, she said to her husband, ". . . Behold now, 
the Lord hath restrained me from bearing: I pray thee, go in unto 
my maid; it may be that I may obtain children by her . . ." 
(Genesis 16:2). 

Similarly, Jacob's wife, Rachel, lent her husband her maid, Bil- 
hah, while his other wife, Leah, not to be behindhand, made her 
maid Zilpah available. These four women, among them, are 
described as being the mothers of the various ancestors of the 
twelve tribes of Israel. 

It worked the other way, too. If a husband died before having 
children, it was the duty of the nearest member of the family 
(the brother, if possible) to make the effort of impregnating the 
widow in order that she might have sons which would then be 
counted to the credit of the dead man. 

Thus, Jacob's fourth son, Judah, had an oldest son, Er, for 
whom he arranged a marriage with a young lady named Tamar. 
Unfortunately, Er died, so Judah told his next son, Onan: 
". . . Go in unto thy brother's wife and marry her, and raise up 
seed to thy brother" (Genesis 38:8). 

Onan, however, did not want to. "And Onan knew that the 
seed should not be his; and it came to pass, when he went in unto 
his brother's wife, that he spilled it on the ground, lest that he 
should give seed to his brother. / And the thing which he did dis- 
pleased the Lord: wherefore he slew him also" (Genesis 38:9- 
10 ). 

Thus, the sin of Onan is not masturbation (which is what the 
word "onanism" means) but what we call "coitus interruptus." 

The pressure to bear children exists because a tribal society 

---BUT HOW? 


would not long survive without converting women into baby 
machines, and the biblical tales reflect this. 

To be sure, there are religious sects which glorify birth control 
— in the form of chastity and virginity — but almost invariably be- 
cause they expect the imminent end of the Earth.* The early 
Christians were among these and to this day chastity is a Christian 
virtue, and virginity is considered a pretty praiseworthy thing. 
Yet, even so, it is taken for granted in our traditional society that 
the greatest fulfillment a woman can possibly experience on Earth 
is that of becoming a wife and mother, that motherhood is of all 
things on Earth the most sacred, that to have many children is 
really a blessing and to have few children, or none, through some 
act of will, is somehow to be selfish. 

The pressures produce important myths about men, too, for to 
have many children seems to be accepted as proving something 
about a man's virility. Even today, the father of triplets or more 
sometimes manages a look of smug modesty before the camera, 
an "oh-it-was-nothing" expression that he thinks befits the sexual 
athlete. (Actually, whatever a man does or does not do has no 
connection at all with multiple births.) 

All these pressures inherited from the dead past exist, then, 
despite the fact that the situation is now no longer what it was 
in tribal days. It is completely and catastrophically the opposite. 
We no longer have an empty Earth, we have a full one. We no 
longer have a short life expectancy, but a long one. We no longer 
have a high infant mortality rate, but a low one. We are no longer 
doubling Earth's population in several millennia, but in several 

Yet when we speak of birth control even today, we still have 
to overcome all the age-old beliefs of the tribal situation. Clearly, 
social pressure can be fought only with social pressure and as an 
example I have sometimes suggested (with a grin, lest I be 
lynched on the spot) that we begin by abolishing Mother's Day 

* Which, in a way, is why the modem population experts are pushing 
for birth control, too, because otherwise they expect the imminent end of 
the Earth. 


and replacing it with Childless Day, in which we honor all the 
adult women without children. 

Social pressure involves more than merely a question of having 
children or not having children. The social pressures that for 
thousands of years have insisted on children, have gone into detail 
to make sure that these children come to pass. They have defi- 
nitely and specifically outlawed the easiest methods of birth con- 
trol, methods which require no equipment, no chemicals, no 
calculations, no particular self-control, methods which, if applied, 
under tribal conditions of yore, would have threatened the tribe 
with extinction. 

So successful has this pressure been that such methods of birth 
control have passed beyond human ken, apparently. At least, when 
I hear proponents of birth control speak, or read what they write, 
I never seem to hear or see any mention of these natural methods. 
Either they are blissfully ignorant of them, or are afraid to speak 
of them. 

The fact is, you see, that there are a variety of sexual practices 
that seem to give satisfaction, that do no physiological harm, and 
that offer no chance, whatsoever, for conception. 

One and all, these stand condemned in our society for reasons 
that stretch back to the primitive necessity for babies. 

For instance, the simplest possible non-conception-centered 
sexual practice is masturbation (in either male or female). It re- 
duces tension and does no physiological harm. 

Yet for how many years in our own society has it been viewed 
as an unspeakable vice (despite the fact that, I understand, it is 
almost universally practiced). The pressure to consider it as more 
than a vice, and as actually a sin, has been such that in the effort 
to find biblical thunder against it, Onan's deed was considered 
masturbation, which it most certainly was not. 

Clearly, the real crime of masturbation is that it wastes semen 
which, by tribal views, ought to be used in a sporting effort to 
effect conception. To say this, however, would be alien to the 
spirit of our society, so lies are invented instead. Masturbation 



(the threat goes) "weakens" you; by which is meant that you 
won't perform effectively with women — a horrifying possibility 
to most men. Worse than that is the wild threat that masturba- 
tion gives rise to degeneracy (whatever that is) and even insanity. 

Actually, it does none of these things. It does not even have 
the evils implicit in its being a "solitary vice." It can be indulged 
in, in company, and not necessarily in "vile orgies," but in 
ordinary heterosexual interaction. 

All the strictures and fulminations against masturbation have 
never succeeded in wiping it out. It continued universal. What 
the lies did do, however, was to force the act to be carried on in 
secret, in shame, and in fear, so that those lies helped raise up 
generations of neurotics with distorted and utterly unnecessary 
hangups about sex. And why? To pay lip service to practices neces- 
sary to primitive tribes, but fatal to ourselves. 

Part and parcel of the battle against masturbation is that 
against pornography. There have been periods in history when 
pornography was driven underground with scorn and disgust. 
This did not wipe out "dirty books," "dirty pictures," and "dirty 
jokes." It lent them an added titillation, if anything. But the drive 
against pornography did make it clear that sex was filthy, and 
therefore utterly distorted the attitude of millions concerning an 
activity which is both necessary and intensely pleasurable. 

And what is the reason usually given for forbidding pornogra- 
phy? The one I hear most often is that it will inflame minds and 
cause people encountering such "filth" to go ravening out into 
the street like wild beasts, seeking to rape and pervert. 

It is ridiculous to think so. I suspect that what happens when 
you involve yourself with pornography, assuming it succeeds in 
arousing "vile impulses" within you, is that you masturbate at 
the first opportunity. It releases tension rather than building it. 

It is, in fact, by building tensions through a studied effort to 
consider sex dirty and forbidden, that one is most likely to be 
driven to rape. 

No, the real evil of pornography in a tribal society is that, by 
encouraging masturbation, it diminishes the chance of conception. 


There is a whole array of practices which, by the society and 
therefore by law, are stigmatized as perverse, as unnatural, as 
unspeakable, as "crimes against nature," and so on. That these 
are unnatural is clearly not so, for if they were they would be 
easy to suppress. Indeed, there would be no need to suppress 
them, for they wouldn't exist. It is unnatural, for instance, to fly 
by flapping your arms, so that there are no laws against it. It is 
unnatural to live without breathing, so no one has to inveigh 
against it. 

What is true about the so-called perversions is that they are 
very natural. They are so natural, indeed, that not all the shackles 
of the law, and not all the hellflre of religion, can serve to wipe 
them out. 

And what harm do they do? Are they sicknesses? 

I frequently hear homosexuality spoken of as a sickness, for 
instance, and yet there have been societies in which it was taken 
more or less for granted. Homosexuality was prevalent, and even 
approved, in the Golden Age of Athens; it was prevalent during 
the Golden Age of Islam; and despite everything, it was prevalent 
(I understand) among the upper classes of the Victorian Age. 

It may be sickness but it does not seem to be inconsistent with 
culture. And how much of its sickness is the result of the hidden 
world in which it is forced to live, the fear and shame that are 
made to accompany it? 

What is the real crime of all these so-called perversions? Might 
it not be that one and all are effective birth control agents. No 
practicing, exclusive homosexual, male or female, can possibly 
make or become pregnant. No one can ever impregnate or be 
impregnated by oral-genital contacts. 

So what's wrong — in a time when birth rate must be lowered? 

I don't mean that there aren't practices that do do harm, and 
these one ought to oppose. Sadomasochistic practices carried be- 
yond the level of mild stimulation are not to be encouraged, for 
the same reason we oppose mutilation and murder. Those prac- 
tices which involve seriously unhygienic conditions should be dis- 



couraged for the same reason any other unhygienic condition is 

Nor do I imply that we must force people to practice perver- 

I, for instance, am not a homosexual and wouldn't consider 
becoming one just to avoid having children. Nor would I persuade 
anyone to become one for that purpose and that purpose only. 

I merely say that in a world threatened by overconception, it 
is useless and even suicidally harmful to carry on a battle against 
those who, of their own accord, prefer homosexuality, who in 
doing so do us no harm, and who, indeed, spare us children. Fur- 
thermore, there are borderline cases who might be homosexuals 
if left to themselves; shall we force them, by unbearable social 
pressure, into loveless heterosexual marriages, and into presenting 
the world with unneeded babies? 

How do we justify this in the endangered world of the late 
twentieth century? 

Social pressure — and the law — invades the bedrooms of even 
legally married individuals and dictates their private sexual prac- 
tices. I am told that if a man and wife wish to practice anal or oral 
intercourse and are caught at it, they can be given stiff jail sen- 
tences in almost any state of the Union. 

Why? What harm have they done themselves or anyone else? 
It is punishment without crime. 

The "harm," of course, is that they've practiced a completely 
effective birth control method that requires no equipment, no 
preparation, and supplies them, presumably, with satisfaction- 
something incompatible with the needs of a long-dead-and-gone 
tribal past. 

I have heard it said that the practice of "perversions" is "cor- 
rupting," that it replaces the "normal way," which is then neg- 

I've never seen evidence presented to back this view, but even 
if it were true, what then? What is the "normal way" in a world 
like ours which must dread overconception? And if someone 
doesn't like the "normal way" and therefore doesn't have children, 




whose business is that? If that same couple chose not to have chil- 
dren by practicing abstention, would anyone care? Would the 
law care? Then what's wrong with not having children another 
way? Because pleasure is a crime? 

In David Reuben's book Everything You Always Wanted to 
Know about Sex, he devotes a section to oral-genital contacts, of 
which he seems to approve, but concludes that "regular copula- 
tion is even more enjoyable." 

Actually, I suppose that is for each individual to decide for him- 
self, but even if "regular copulation" is more enjoyable, what 
then? If you find roast beef more enjoyable than bread and butter, 
is that a reason to outlaw bread and butter? And if you can't have 
roast beef and must choose between bread and butter and starva- 
tion, would you choose starvation? 

It might very well be that it is variety that is best of all, and 
that for law and custom to try to insist on a monotony which, of 
all monotonies, is most dangerous to us today, is the greatest per- 
version of all. 

Let's summarize, then. 

I think that the importance of birth control is such that we 
ought to allow no useful method to lie unused. 

All the common methods have their drawbacks: abstention is 
nearly impossible; sterilization is abhorrent; the rhythm method 
is cold-blooded and deprives the female of sex at just the time of 
the month she is most receptive; mechanical devices slow you up 
just when you least want to slow up; chemicals are bound to have 
side effects. I think, then, there is room for another method, par- 
ticularly one which has none of these drawbacks. 

With that in mind I think that social pressure against those 
practices commonly called "perversions" ought to be lifted, where 
these are not physiologically harmful. The very qualities that 
made them perversions in a conception-centered society make 
them virtues in a non-conception-centered society. 

I think that sex education ought to include not only informa- 
tion concerning what is usually considered "normal" but also 



about those practices which are non-conception-centered. No one 
need to be taught to indulge in them exclusively, but by knowing 
they exist and aren't "wrong," the number of occasions that so- 
called normal intercourse need be indulged in, with all the com- 
plications and drawbacks of artificial birth control methods, can 
be reduced. And, of course, if a couple have no children, and want 
one or two, they will know what to do. 

As for those who can't stomach "perversions" and who insist 
on doing everything by the numbers in the way that was good 
enough for their grandmother (I wonder!), then good luck to 
them, but they had better be careful. 

One way or another, birth control must be made effective, and 
what I have suggested here is only one more method; one which, 
joined to the others already available, increases by that much the 
general effectiveness of the system as a whole and makes the 
chance just a little bit greater that the world might yet be saved.